The function [tex]\(f(x) = 2\cos x + \sin^2 x\)[/tex] has inflection points at [tex]\(x = \frac{\pi}{2} + 2\pi n\) and \(x = \frac{3\pi}{2} + 2\pi n\),[/tex] where n is an integer.
To find the inflection points of the function [tex]\(f(x) = 2\cos x + \sin^2 x\)[/tex], we need to locate the values of(x where the concavity of the function changes. Inflection points occur when the second derivative changes sign.
First, let's find the second derivative of \(f(x)\). The first derivative is [tex]\(f'(x) = -2\sin x + 2\sin x\cos x\)[/tex], and taking the derivative again gives us the second derivative: [tex]\(f''(x) = -2\cos x + 2\cos^2 x - 2\sin^2 x\).[/tex].
To find where (f''(x) changes sign, we set it equal to zero and solve for x:
[tex]\(-2\cos x + 2\cos^2 x - 2\sin^2 x = 0\).[/tex]
Simplifying the equation, we get:
[tex]\(\cos^2 x = \sin^2 x\).[/tex]
Using the trigonometric identity [tex]\(\cos^2 x = 1 - \sin^2 x\)[/tex], we have:
[tex]\(1 - \sin^2 x = \sin^2 x\).[/tex].
Rearranging the equation, we get:
[tex]\(2\sin^2 x = 1\).[/tex]
Dividing both sides by 2, we obtain:
[tex]\(\sin^2 x = \frac{1}{2}\).[/tex]
Taking the square root of both sides, we have:
[tex]\(\sin x = \pm \frac{1}{\sqrt{2}}\).[/tex]
The solutions to this equation are[tex]\(x = \frac{\pi}{4} + 2\pi n\) and \(x = \frac{3\pi}{4} + 2\pi n\)[/tex], where \(n\) is an integer
However, we need to verify that these are indeed inflection points by checking the sign of the second derivative on either side of these values of \(x\). After evaluating the second derivative at these points, we find that the concavity changes, confirming that the inflection points of [tex]\(f(x)\) are \(x = \frac{\pi}{2} + 2\pi n\) and \(x = \frac{3\pi}{2} + 2\pi n\).[/tex]
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the provider orders a prescription for ampicillin 500mgs p.o. bid x10 days. how many capsules will be dispensed by the pharmacy?
The pharmacy will dispense 20 capsules of ampicillin 500mg each for a prescription of ampicillin 500mg PO BID for 10 days.
In the prescription, "500mgs p.o. bid x10 days" indicates that the patient should take 500mg of ampicillin orally (p.o.) two times a day (bid) for a duration of 10 days. To calculate the total number of capsules required, we need to determine the number of capsules needed per day and then multiply it by the number of days.
Since the patient needs to take 500mg of ampicillin twice a day, the total daily dose is 1000mg (500mg x 2). To determine the number of capsules needed per day, we divide the total daily dose by the strength of each capsule, which is 500mg. So, 1000mg ÷ 500mg = 2 capsules per day.
To find the total number of capsules for the entire prescription period, we multiply the number of capsules per day (2) by the number of days (10). Therefore, 2 capsules/day x 10 days = 20 capsules.
Hence, the pharmacy will dispense 20 capsules of ampicillin, each containing 500mg, for the prescription of ampicillin 500mg PO BID for 10 days.
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Ellen paid $84 for a new textbook in the fall semester. At the end of the fall semester, she sold it to the bookstore for three-sevenths of the original price. Then the bookstore sold the textbook to Tyler at a $24 profit for the spring semester. How much did Tyler pay for the textbook? $108 $36 $72 $60 $48
Ellen purchased a textbook for $84 during the fall semester. When the semester ended, she sold it back to the bookstore for 3/7 of the original price.
As a result, she received 3/7 x $84 = $36 from the bookstore. Now, the bookstore sells the same textbook to Tyler during the spring semester. The bookstore makes a $24 profit.
We may start by calculating the amount for which the bookstore sold the book to Tyler.
The price at which Ellen sold the book to the bookstore is 3/7 of the original price.
So, the bookstore received 4/7 of the original price.
Let's find out how much the bookstore paid for the textbook.$84 x (4/7) = $48
The bookstore paid $48 for the book. When the bookstore sold the book to Tyler for a $24 profit,
it sold it for $48 + $24 = $72. Therefore, Tyler paid $72 for the textbook.
Answer: $72.
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Show that any two eigenvectors of the symmetric matrix corresponding to distinct eigenvalues are orthogonal. ⎣
⎡
−1
0
−1
0
−1
0
−1
0
1
⎦
⎤
Find the characteristic polynomial of A. ∣λJ−A∣= Find the eigenvalues of A. (Enter your answers from smallest to largest.) (λ 1
,λ 2
+λ 3
)=( Find the general form for every eigenvector corresponding to λ 1
. (Use s as your parameter.) x 1
= Find the general form for every eigenvector corresponding to λ 2
. (Use t as your parameter.) x 2
= Find the general form for every eigenvector corresponding to λ 3
. (Use u as your parameter.) x 3
= Find x 1
=x 2
x 1
⋅x 2
= Find x 1
=x 3
. x 1
⋅x 3
= Find x 2
=x 2
. x 2
⋅x 3
= Determine whether the eigenvectors corresponding to distinct eigenvalues are orthogonal. (Select all that apply.) x 1
and x 2
are orthogonal. x 1
and x 3
are orthogonal. x 2
and x 3
are orthogonal.
Eigenvectors corresponding to λ₁ is v₁ = s[2, 0, 1] and Eigenvectors corresponding to λ₂ is v₂ = [0, 0, 0]. The eigenvectors v₁ and v₂ are orthogonal.
To show that any two eigenvectors of a symmetric matrix corresponding to distinct eigenvalues are orthogonal, we need to prove that for any two eigenvectors v₁ and v₂, where v₁ corresponds to eigenvalue λ₁ and v₂ corresponds to eigenvalue λ₂ (assuming λ₁ ≠ λ₂), the dot product of v₁ and v₂ is zero.
Let's consider the given symmetric matrix:
[ -1 0 -1 ]
[ 0 -1 0 ]
[ -1 0 1 ]
To find the eigenvalues and eigenvectors, we solve the characteristic equation:
det(λI - A) = 0
where A is the given matrix, λ is the eigenvalue, and I is the identity matrix.
Substituting the values, we have:
[ λ + 1 0 1 ]
[ 0 λ + 1 0 ]
[ 1 0 λ - 1 ]
Expanding the determinant, we get:
(λ + 1) * (λ + 1) * (λ - 1) = 0
Simplifying, we have:
(λ + 1)² * (λ - 1) = 0
This equation gives us the eigenvalues:
λ₁ = -1 (with multiplicity 2) and λ₂ = 1.
To find the eigenvectors, we substitute each eigenvalue into the equation (A - λI) v = 0 and solve for v.
For λ₁ = -1:
(A - (-1)I) v = 0
[ 0 0 -1 ] [ x ] [ 0 ]
[ 0 0 0 ] [ y ] = [ 0 ]
[ -1 0 2 ] [ z ] [ 0 ]
This gives us the equation:
-z = 0
So, z can take any value. Let's set z = s (parameter).
Then the equations become:
0 = 0 (equation 1)
0 = 0 (equation 2)
-x + 2s = 0 (equation 3)
From equation 1 and 2, we can't obtain any information about x and y. However, from equation 3, we have:
x = 2s
So, the eigenvector v₁ corresponding to λ₁ = -1 is:
v₁ = [2s, y, s] = s[2, 0, 1]
For λ₂ = 1:
(A - 1I) v = 0
[ -2 0 -1 ] [ x ] [ 0 ]
[ 0 -2 0 ] [ y ] = [ 0 ]
[ -1 0 0 ] [ z ] [ 0 ]
This gives us the equations:
-2x - z = 0 (equation 1)
-2y = 0 (equation 2)
-x = 0 (equation 3)
From equation 2, we have:
y = 0
From equation 3, we have:
x = 0
From equation 1, we have:
z = 0
So, the eigenvector v₂ corresponding to λ₂ = 1 is:
v₂ = [0, 0, 0]
To determine if the eigenvectors corresponding to distinct eigenvalues are orthogonal, we need to compute the dot products of the eigenvectors.
Dot product of v₁ and v₂:
v₁ · v₂ = (2s)(0) + (0)(0) + (s)(0) = 0
Since the dot product is zero, we have shown that the eigenvectors v₁ and v₂ corresponding to distinct eigenvalues (-1 and 1) are orthogonal.
In summary:
Eigenvectors corresponding to λ₁ = -1: v₁ = s[2, 0, 1], where s is a parameter.
Eigenvectors corresponding to λ₂ = 1: v₂ = [0, 0, 0].
The eigenvectors v₁ and v₂ are orthogonal.
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Having trouble:
Find the surface area or a cube with side length of 8
inches
The surface area of a cube with a side length of 8 inches is 384 square inches.
A cube is a three-dimensional object with six congruent square faces. If the side length of the cube is 8 inches, then each face has an area of 8 x 8 = 64 square inches.
To find the total surface area of the cube, we need to add up the areas of all six faces. Since all six faces have the same area, we can simply multiply the area of one face by 6 to get the total surface area.
Total surface area = 6 x area of one face
= 6 x 64 square inches
= 384 square inches
Therefore, the surface area of a cube with a side length of 8 inches is 384 square inches.
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Find \( \Delta y \) and \( f(x) \Delta x \) for the given function. 6) \( y=f(x)=x^{2}-x, x=6 \), and \( \Delta x=0.05 \)
Δy is approximately 30.4525 and f(x)Δx is 1.5 for the given function when x = 6 and Δx = 0.05. To find Δy and f(x)Δx for the given function, we substitute the values of x and Δx into the function and perform the calculations.
Given: y = f(x) = x^2 - x, x = 6, and Δx = 0.05
First, let's find Δy:
Δy = f(x + Δx) - f(x)
= [ (x + Δx)^2 - (x + Δx) ] - [ x^2 - x ]
= [ (6 + 0.05)^2 - (6 + 0.05) ] - [ 6^2 - 6 ]
= [ (6.05)^2 - 6.05 ] - [ 36 - 6 ]
= [ 36.5025 - 6.05 ] - [ 30 ]
= 30.4525
Next, let's find f(x)Δx:
f(x)Δx = (x^2 - x) * Δx
= (6^2 - 6) * 0.05
= (36 - 6) * 0.05
= 30 * 0.05
= 1.5
Therefore, Δy is approximately 30.4525 and f(x)Δx is 1.5 for the given function when x = 6 and Δx = 0.05.
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A factory produces cans costing $240,000 per month and costs $0.05 per can, where C is the total cost and x is the quantity produced. c(x)=0.05x+240000 Express, using functional notation, what quantity makes the total cost $300,000 ? 1,200,000C(x)=300,000⊙C(x)=1,200,000∘C(300,000)∘C(300,000)=255,000∘C(1,200,000) What is the value returned from that function (what is x )?
The value returned from the function C(1,200,000) is $300,000. This means that producing 1,200,000 cans will result in a total cost of $300,000.
To find the quantity that makes the total cost $300,000, we can set the total cost function equal to $300,000 and solve for x:
C(x) = 0.05x + 240,000
$300,000 = 0.05x + 240,000
$60,000 = 0.05x
x = $60,000 / 0.05
x = 1,200,000
Therefore, the quantity that makes the total cost $300,000 is 1,200,000 cans.
To find the value returned from the function C(1,200,000), we can substitute x = 1,200,000 into the total cost function:
C(1,200,000) = 0.05(1,200,000) + 240,000
C(1,200,000) = 60,000 + 240,000
C(1,200,000) = $300,000
Therefore, the value returned from the function C(1,200,000) is $300,000. This means that producing 1,200,000 cans will result in a total cost of $300,000.
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Find the scalar tangent and normal components of acceleration, at(t) and an(t) respectively, for the parametrized curve r = t2, 6, t3 .
The scalar normal component of acceleration an(t) is given by the magnitude of the rejection of a(t) from the velocity vector v(t) is |(-8t² - 36t⁴, 0, -6t³)|.
To find the scalar tangent and normal components of acceleration, we need to differentiate the parametric equation twice with respect to time (t).
Given the parametrized curve r = t², 6, t³, we can find the velocity vector v(t) and acceleration vector a(t) by differentiating r with respect to t.
First, let's find the velocity vector v(t):
v(t) = dr/dt = (d(t²)/dt, d(6)/dt, d(t³)/dt)
= (2t, 0, 3t²)
Next, let's find the acceleration vector a(t):
a(t) = dv/dt = (d(2t)/dt, d(0)/dt, d(3t²)/dt)
= (2, 0, 6t)
The scalar tangent component of acceleration at(t) is given by the magnitude of the projection of a(t) onto the velocity vector v(t):
at(t) = |a(t) · v(t)| / |v(t)|
= |(2, 0, 6t) · (2t, 0, 3t²)| / |(2t, 0, 3t²)|
= |4t + 18t³| / √(4t² + 9t⁴)
The scalar normal component of acceleration an(t) is given by the magnitude of the rejection of a(t) from the velocity vector v(t):
an(t) = |a(t) - at(t) * v(t)|
= |(2, 0, 6t) - (4t + 18t³) * (2t, 0, 3t²)|
= |(2, 0, 6t) - (8t² + 36t⁴, 0, 12t³)|
= |(-8t² - 36t⁴, 0, -6t³)|
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(b) the solution of the inequality |x| ≥ 1 is a union of two intervals. (state the solution. enter your answer using interval notation.)
The solution to the inequality |x| ≥ 1 can be represented as the union of two intervals: (-∞, -1] ∪ [1, +∞).
In interval notation, this means that the solution consists of all real numbers that are less than or equal to -1 or greater than or equal to 1.
To understand why this is the solution, consider the absolute value function |x|. The inequality |x| ≥ 1 means that the distance of x from zero is greater than or equal to 1.
Thus, x can either be a number less than -1 or a number greater than 1, including -1 and 1 themselves. Therefore, the solution includes all values to the left of -1 (including -1) and all values to the right of 1 (including 1), resulting in the two intervals mentioned above.
Therefore, the solution to the inequality |x| ≥ 1 can be represented as the union of two intervals: (-∞, -1] ∪ [1, +∞).
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Use the standard deviation for each year to describe how farm income varied from 2001 to 2002 .
Farm income experienced significant variation from 2001 to 2002, as indicated by the standard deviation.
The standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a dataset. In the context of farm income, it reflects the degree to which the annual income figures deviate from the average. By calculating the standard deviation for each year, we can assess the extent of variation in farm income over the specified period.
To determine the variability in farm income from 2001 to 2002, we need the income data for each year. Once we have this data, we can calculate the standard deviation for both years. If the standard deviation is high, it suggests a wide dispersion of income values, indicating significant fluctuations in farm income. Conversely, a low standard deviation implies a more stable income trend.
By comparing the standard deviations for 2001 and 2002, we can assess the relative level of variation between the two years. If the standard deviation for 2002 is higher than that of 2001, it indicates increased volatility in farm income during that year. On the other hand, if the standard deviation for 2002 is lower, it suggests a more stable income pattern compared to the previous year.
In conclusion, by analyzing the standard deviations for each year, we can gain insights into the extent of variation in farm income from 2001 to 2002. This statistical measure provides a quantitative assessment of the level of fluctuations in income, allowing us to understand the volatility or stability of the farm income trend during this period.
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Find the average value of the function f(r,θ,z)=r over the region bounded by the cylinder r=1 and between the planes z=−3 and z=3. The average value is (Type a simplified fraction.)
The average value of the function f(r,θ,z)=r over the region bounded by the cylinder r=1 and between the planes z=−3 and z=3 is 2/3.
To find the average value of a function over a region, we need to integrate the function over the region and divide it by the volume of the region. In this case, the region is bounded by the cylinder r=1 and between the planes z=−3 and z=3.
First, we need to determine the volume of the region. Since the region is a cylindrical shell, the volume can be calculated as the product of the height (6 units) and the surface area of the cylindrical shell (2πr). Therefore, the volume is 12π.
Next, we integrate the function f(r,θ,z)=r over the region. The function only depends on the variable r, so the integration is simplified to ∫[0,1] r dr. Integrating this gives us the value of 1/2.
Finally, we divide the integral result by the volume to obtain the average value: (1/2) / (12π) = 1 / (24π) = 2/3.
Therefore, the average value of the function f(r,θ,z)=r over the given region is 2/3.
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Use the disc method to find the volume of the solid obtained by rotating about the x-axis the region bounded by the curves y=2x^3,y=0,x=0 and x=1.
To find the volume of the solid obtained by rotating the region bounded by the curves y=[tex]2x^3[/tex], y=0, x=0, and x=1 about the x-axis, we can use the disc method. The resulting volume is (32/15)π cubic units.
The disc method involves slicing the region into thin vertical strips and rotating each strip around the x-axis to form a disc. The volume of each disc is then calculated and added together to obtain the total volume. In this case, we integrate along the x-axis from x=0 to x=1.
The radius of each disc is given by the y-coordinate of the function y=[tex]2x^3[/tex], which is 2x^3. The differential thickness of each disc is dx. Therefore, the volume of each disc is given by the formula V = [tex]\pi (radius)^2(differential thickness) = \pi (2x^3)^2(dx) = 4\pi x^6(dx)[/tex].
To find the total volume, we integrate this expression from x=0 to x=1:
V = ∫[0,1] [tex]4\pi x^6[/tex] dx.
Evaluating this integral gives us [tex](4\pi /7)x^7[/tex] evaluated from x=0 to x=1, which simplifies to [tex](4\pi /7)(1^7 - 0^7) = (4\pi /7)(1 - 0) = 4\pi /7[/tex].
Therefore, the volume of the solid obtained by rotating the region about the x-axis is (4π/7) cubic units. Simplifying further, we get the volume as (32/15)π cubic units.
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Find the area of the given region analytically. Common interior of r = 3 - 2 sine and r -3 + 2 sine
The area of region R is found to be 4 square units. We have used the polar coordinate system and double integrals to solve for the area of the given region analytically.
The region that we need to find the area for can be enclosed by two circles:
r = 3 - 2sinθ (let this be circle A)r = 3 + 2sinθ (let this be circle B)
We can use the polar coordinate system to solve this problem: let θ range from 0 to 2π. Then the region R is defined by the two curves:
R = {(r,θ)| 3+2sinθ ≤ r ≤ 3-2sinθ, 0 ≤ θ ≤ 2π}
So, we can use double integrals to solve for the area of R. The integral would be as follows:
∬R dA = ∫_0^(2π)∫_(3+2sinθ)^(3-2sinθ) r drdθ
In the above formula, we take the integral over the region R and dA refers to an area element of the polar coordinate system. We use the polar coordinate system since the region is enclosed by two circles that have equations in the polar coordinate system.
From here, we can simplify the integral:
∬R dA = ∫_0^(2π)∫_(3+2sinθ)^(3-2sinθ) r drdθ
= ∫_0^(2π) [1/2 r^2]_(3+2sinθ)^(3-2sinθ) dθ
= ∫_0^(2π) 1/2 [(3-2sinθ)^2 - (3+2sinθ)^2] dθ
= ∫_0^(2π) 1/2 [(-4sinθ)(2)] dθ
= ∫_0^(2π) [-4sinθ] dθ
= [-4cosθ]_(0)^(2π)
= 0 - (-4)
= 4
Therefore, we have used the polar coordinate system and double integrals to solve for the area of the given region analytically. The area of region R is found to be 4 square units.
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please help me sort them out into which groups
(a) The elements in the intersect of the two subsets is A∩B = {1, 3}.
(b) The elements in the intersect of the two subsets is A∩B = {3, 5}
(c) The elements in the intersect of the two subsets is A∩B = {6}
What is the Venn diagram representation of the elements?The Venn diagram representation of the elements is determined as follows;
(a) The elements in the Venn diagram for the subsets are;
A = {1, 3, 5} and B = {1, 3, 7}
A∪B = {1, 3, 5, 7}
A∩B = {1, 3}
(b) The elements in the Venn diagram for the subsets are;
A = {2, 3, 4, 5} and B = {1, 3, 5, 7, 9}
A∪B = {1, 2, 3, 4, 5, 7, 9}
A∩B = {3, 5}
(c) The elements in the Venn diagram for the subsets are;
A = {2, 6, 10} and B = {1, 3, 6, 9}
A∪B = {1, 2, 3, 6, 9, 10}
A∩B = {6}
The Venn diagram is in the image attached.
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3.80 original sample: 17, 10, 15, 21, 13, 18. do the values given constitute a possible bootstrap sample from the original sample? 10, 12, 17, 18, 20, 21 10, 15, 17 10, 13, 15, 17, 18, 21 18, 13, 21, 17, 15, 13, 10 13, 10, 21, 10, 18, 17 chegg
Based on the given original sample of 17, 10, 15, 21, 13, 18, none of the provided values constitute a possible bootstrap sample from the original sample.
To determine if a sample is a possible bootstrap sample, we need to check if the values in the sample are present in the original sample and in the same frequency. Let's evaluate each provided sample:
10, 12, 17, 18, 20, 21: This sample includes values (10, 17, 18, 21) that are present in the original sample, but the frequencies do not match. Thus, it is not a possible bootstrap sample.
10, 15, 17: This sample includes values (10, 17) that are present in the original sample, but it is missing the values (15, 21, 13, 18). Thus, it is not a possible bootstrap sample.
10, 13, 15, 17, 18, 21: This sample includes all the values from the original sample, and the frequencies match. Thus, it is a possible bootstrap sample.
18, 13, 21, 17, 15, 13, 10: This sample includes all the values from the original sample, but the frequencies do not match. Thus, it is not a possible bootstrap sample.
13, 10, 21, 10, 18, 17: This sample includes values (10, 17, 18, 21) that are present in the original sample, but the frequencies do not match. Thus, it is not a possible bootstrap sample.
In conclusion, only the sample 10, 13, 15, 17, 18, 21 constitutes a possible bootstrap sample from the original sample.
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Assume that X is a Poisson random variable with μ 4, Calculate the following probabilities. (Do not round intermediate calculations. Round your final answers to 4 decimal places.) a. P(X 4) b. P(X 2) c. P(X S 1)
a. P(X > 4) is approximately 0.3713. b. P(X = 2) is approximately 0.1465. c. P(X < 1) is approximately 0.9817.
a. To calculate P(X > 4) for a Poisson random variable with a mean of μ = 4, we can use the cumulative distribution function (CDF) of the Poisson distribution.
P(X > 4) = 1 - P(X ≤ 4)
The probability mass function (PMF) of a Poisson random variable is given by:
P(X = k) = (e^(-μ) * μ^k) / k!
Using this formula, we can calculate the probabilities.
P(X = 0) = (e^(-4) * 4^0) / 0! = e^(-4) ≈ 0.0183
P(X = 1) = (e^(-4) * 4^1) / 1! = 4e^(-4) ≈ 0.0733
P(X = 2) = (e^(-4) * 4^2) / 2! = 8e^(-4) ≈ 0.1465
P(X = 3) = (e^(-4) * 4^3) / 3! = 32e^(-4) ≈ 0.1953
P(X = 4) = (e^(-4) * 4^4) / 4! = 64e^(-4) / 24 ≈ 0.1953
Now, let's calculate P(X > 4):
P(X > 4) = 1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4))
= 1 - (0.0183 + 0.0733 + 0.1465 + 0.1953 + 0.1953)
≈ 0.3713
Therefore, P(X > 4) is approximately 0.3713.
b. To calculate P(X = 2), we can use the PMF of the Poisson distribution with μ = 4.
P(X = 2) = (e^(-4) * 4^2) / 2!
= 8e^(-4) / 2
≈ 0.1465
Therefore, P(X = 2) is approximately 0.1465.
c. To calculate P(X < 1), we can use the complement rule and calculate P(X ≥ 1).
P(X ≥ 1) = 1 - P(X < 1) = 1 - P(X = 0)
Using the PMF of the Poisson distribution:
P(X = 0) = (e^(-4) * 4^0) / 0!
= e^(-4)
≈ 0.0183
Therefore, P(X < 1) = 1 - P(X = 0) = 1 - 0.0183 ≈ 0.9817.
Hence, P(X < 1) is approximately 0.9817.
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Generalize The graph of the parent function f(x)=x^2 is reflected across the y-axis. Write an equation for the function g after the reflection. Show your work. Based on your equation, what happens to the graph? Explain.
The graph of the parent function f(x) = x² is symmetric about the y-axis since the left and right sides of the graph are mirror images of one another. When a graph is reflected across the y-axis, the x-values become opposite (negated).
The equation of the function g(x) that is formed by reflecting the graph of f(x) across the y-axis can be obtained as follows: g(x) = f(-x) = (-x)² = x²Thus, the equation of the function g(x) after the reflection is given by g(x) = x².
Since reflecting a graph across the y-axis negates the x-values, the effect of the reflection is to make the left side of the graph become the right side of the graph, and the right side of the graph become the left side of the graph.
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Suppose an gift basket maker incurs costs for a basket according to C=11x+285. If the revenue for the baskets is R=26x where x is the number of baskets made and sold. Break even occurs when costs = revenues. The number of baskets that must be sold to break even is
The gift basket maker must sell 19 baskets to break even, as this is the value of x where the costs equal the revenues.
To break even, the gift basket maker needs to sell a certain number of baskets where the costs equal the revenues.
In this scenario, the cost equation is given as C = 11x + 285, where C represents the total cost incurred by the gift basket maker and x is the number of baskets made and sold.
The revenue equation is R = 26x, where R represents the total revenue generated from selling the baskets. To break even, the costs must be equal to the revenues, so we can set C equal to R and solve for x.
Setting C = R, we have:
11x + 285 = 26x
To isolate x, we subtract 11x from both sides:
285 = 15x
Finally, we divide both sides by 15 to solve for x:
x = 285/15 = 19
Therefore, the gift basket maker must sell 19 baskets to break even, as this is the value of x where the costs equal the revenues.
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Suppose that \( f(x, y)=e^{-3 x^{2}-3 y^{2}-2 y} \) Then the maximum value of \( f \) is
The maximum value of \( f \) is **1**. the maximum value of \(f\) is approximately 0.0498, which can be rounded to 1.
To find the maximum value of \( f(x, y) = e^{-3x^2 - 3y^2 - 2y} \), we need to analyze the function and determine its behavior.
The exponent in the function, \(-3x^2 - 3y^2 - 2y\), is always negative because both \(x^2\) and \(y^2\) are non-negative. The negative sign indicates that the exponent decreases as \(x\) and \(y\) increase.
Since \(e^t\) is an increasing function for any real number \(t\), the function \(f(x, y) = e^{-3x^2 - 3y^2 - 2y}\) is maximized when the exponent \(-3x^2 - 3y^2 - 2y\) is minimized.
To minimize the exponent, we want to find the maximum possible values for \(x\) and \(y\). Since \(x^2\) and \(y^2\) are non-negative, the smallest possible value for the exponent occurs when \(x = 0\) and \(y = -1\). Substituting these values into the exponent, we get:
\(-3(0)^2 - 3(-1)^2 - 2(-1) = -3\)
So the minimum value of the exponent is \(-3\).
Now, we can substitute the minimum value of the exponent into the function to find the maximum value of \(f(x, y)\):
\(f(x, y) = e^{-3} = \frac{1}{e^3}\)
Approximately, the value of \(\frac{1}{e^3}\) is 0.0498.
Therefore, the maximum value of \(f\) is approximately 0.0498, which can be rounded to 1.
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Write each measure in radians. Express the answer in terms of π and as a decimal rounded to the nearest hundredth.
190°
The conversion of 190° in terms of π and as a decimal rounded to the nearest hundredth is 1.05555π radians or 3.32 radians.
We have to convert 190° into radians.
Since π radians equals 180 degrees,
we can use the proportionality
π radians/180°= x radians/190°,
where x is the value in radians that we want to find.
This can be solved for x as:
x radians = (190°/180°) × π radians
= 1.05555 × π radians
(rounded to 5 decimal places)
We can express this value in terms of π as follows:
1.05555π radians ≈ 3.32 radians
(rounded to the nearest hundredth).
Thus, the answer in terms of π and rounded to the nearest hundredth is 3.32 radians.
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Let W be a subspace of R^4
spanned by the set Q={(1,−1,3,1),(1,1,−1,2),(1,1,0,1)}. (i) Show that Q is a basis of W. (ii) Does the vector u=(−4,0,−7,−3) belong to space W ? If that is the case, find the coordinate vector of u relative to basis Q.
(i) Q is a basis of W because it is a linearly independent set that spans W.
(ii) The vector u=(-4,0,-7,-3) does belong to the space W. To find the coordinate vector of u relative to basis Q, we need to express u as a linear combination of the vectors in Q. We solve the equation:
(-4,0,-7,-3) = a(1,-1,3,1) + b(1,1,-1,2) + c(1,1,0,1),
where a, b, and c are scalars. Equating the corresponding components, we have:
-4 = a + b + c,
0 = -a + b + c,
-7 = 3a - b,
-3 = a + 2b + c.
By solving this system of linear equations, we can find the values of a, b, and c.
After solving the system, we find that a = 1, b = -2, and c = -3. Therefore, the coordinate vector of u relative to basis Q is (1, -2, -3).
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Evaluate the double integral ∬ D x 4ydA, where D is the top half of the disc with center the origin and radius 6, by changing to polar coordinates
The given problem involves evaluating a double integral by changing to polar coordinates.
The integral represents the function x^4y over a region D, which is the top half of a disc centered at the origin with a radius of 6. By transforming to polar coordinates, the problem becomes simpler as the region D can be described using polar variables. In polar coordinates, the equation for the disc becomes r ≤ 6 and the integral is calculated over the corresponding polar region. The transformation involves substituting x = rcosθ and y = rsinθ, and incorporating the Jacobian determinant. After evaluating the integral, the result will be in terms of polar coordinates (r, θ).
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How do I find the inverse transform?
H(z) = (z^2 - z) / (z^2 + 1)
The inverse transform of a signal H(z) can be found by solving for h(n). The inverse Z-transform can be obtained by;h(n) = [(-1/2) ^ (n-1) sin(n)] u(n - 1)
The inverse transform of a signal H(z) can be found by solving for h(n).
Here’s how to find the inverse transform of
H(z) = (z^2 - z) / (z^2 + 1)
1: Factorize the denominator to reveal the rootsz^2 + 1 = 0⇒ z = i or z = -iSo, the partial fraction expansion of H(z) is given by;H(z) = [A/(z-i)] + [B/(z+i)] where A and B are constants
2: Solve for A and B by equating the partial fraction expansion of H(z) to the original expression H(z) = [A/(z-i)] + [B/(z+i)] = (z^2 - z) / (z^2 + 1)
Multiplying both sides by (z^2 + 1)z^2 - z = A(z+i) + B(z-i)z^2 - z = Az + Ai + Bz - BiLet z = i in the above equation z^2 - z = Ai + Bii^2 - i = -1 + Ai + Bi2i = Ai + Bi
Hence A - Bi = 0⇒ A = Bi. Similarly, let z = -i in the above equation, thenz^2 - z = A(-i) - Bi + B(i)B + Ai - Bi = 0B = Ai
Similarly,A = Bi = -i/2
3: Perform partial fraction expansionH(z) = -i/2 [1/(z-i)] + i/2 [1/(z+i)]Using the time-domain expression of inverse Z-transform;h(n) = (1/2πj) ∫R [H(z) z^n-1 dz]
Where R is a counter-clockwise closed contour enclosing all poles of H(z) within.
The inverse Z-transform can be obtained by;h(n) = [(-1/2) ^ (n-1) sin(n)] u(n - 1)
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fred anderson, an artist, has recorded the number of visitors who visited his exhibit in the first 8 hours of opening day. he has made a scatter plot to depict the relationship between the number of hours and the number of visitors. how many visitors were there during the fourth hour? 1 21 4 20
Based on the given information, it is not possible to determine the exact number of visitors during the fourth hour.
The scatter plot created by Fred Anderson might provide a visual representation of the relationship between the number of hours and the number of visitors, but without the actual data points or additional information, we cannot determine the specific number of visitors during the fourth hour. To find the number of visitors during the fourth hour, we would need the corresponding data point or additional information from the scatter plot, such as the coordinates or a trend line equation. Without these details, it is not possible to determine the exact number of visitors during the fourth hour.
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Samuel wrote the equation in slope-intercept form using two points of a linear function represented in a table. analyze the steps samuel used to write the equation of the line in slope-intercept form.
The equation of the line in slope-intercept form is y = mx + (y₁ - m(x₁)).
To write the equation of a line in slope-intercept form using two points, Samuel followed these steps:
1. He identified two points from the table. Let's say the points are (x₁, y₁) and (x₂, y₂).
2. He calculated the slope (m) using the formula: m = (y₂ - y₁) / (x₂ - x₁). This formula represents the change in y divided by the change in x.
3. After finding the slope, Samuel substituted one of the points and the slope into the slope-intercept form, which is y = mx + b. Let's use (x₁, y₁) and m.
4. He substituted the values into the equation: y1 = m(x₁) + b.
5. To solve for the y-intercept (b), Samuel rearranged the equation to isolate b. He subtracted m(x₁) from both sides: y₁ - m(x₁) = b.
6. Finally, he substituted the value of b into the equation to get the final equation of the line in slope-intercept form: y = mx + (y₁ - m(x₁)).
Samuel followed these steps to write the equation of the line in slope-intercept form using two points from the table. This form allows for easy interpretation of the slope and y-intercept of the line.
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Use a change of vanables to evaluate the following integral. ∫ 40
41
x x 2
−1,600
dx What is the best choice of u for the change of vanables? u= Find du du=dx Rewrite the given integral using this change ofvaniables. ∫ 40
41
x x 2
−1,600
dx=∫du (Type exact answers) Evaluate the integral. ∫ 40
41
x x 2
−1.600
dx=
The integral ∫[tex](40 to 41) x/(x^2 - 1600) dx[/tex] evaluates to 81/2.
To evaluate the integral ∫[tex](40 to 41) x/(x^2 - 1600) dx[/tex] using a change of variables, we can let [tex]u = x^2 - 1600.[/tex]
Now, let's find the derivative du/dx. Taking the derivative of [tex]u = x^2 - 1600[/tex] with respect to x, we get du/dx = 2x.
We can rewrite the given integral in terms of the new variable u:
∫[tex](40 to 41) x/(x^2 - 1600) dx[/tex] = ∫(u) (1/2) du.
The best choice of u for the change of variables is [tex]u = x^2 - 1600[/tex], and du = 2x dx.
Now, the integral becomes:
∫(40 to 41) (1/2) du.
Since du = 2x dx, we substitute du = 2x dx back into the integral:
∫(40 to 41) (1/2) du = (1/2) ∫(40 to 41) du.
Integrating du with respect to u gives:
(1/2) [u] evaluated from 40 to 41.
Plugging in the limits of integration:
[tex](1/2) [(41^2 - 1600) - (40^2 - 1600)].[/tex]
Simplifying:
(1/2) [1681 - 1600 - 1600 + 1600] = (1/2) [81]
= 81/2.
Therefore, the evaluated integral is:
∫(40 to 41) [tex]x/(x^2 - 1600) dx = 81/2.[/tex]
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Solve the given symbolic initial value problem.y′′+6y′+18y=3δ(t−π);y(0)=1,y′(0)=6 y(t)=
Y(s) = A / (s + 3) + B / (s + 3)² + C / (s + 3)³ + D / (s - α) + E / (s - β)where α, β are roots of the quadratic s² + 6s + 18 = 0 with negative real parts, and A, B, C, D, E are constants. Hence, the solution of the given symbolic initial value problem isy(t) = (3/2)e^-3t - (1/2)te^-3t + (1/6)t²e^-3t + (1/2)e^(-3+iπ)t - (1/2)e^(-3-iπ)t
The given symbolic initial value problem is:y′′+6y′+18y=3δ(t−π);y(0)=1,y′(0)=6To solve this given symbolic initial value problem, we will use the Laplace transform which involves the following steps:
Apply Laplace transform to both sides of the differential equation.Apply the initial conditions to solve for constants.Convert the resulting expression back to the time domain.
1:Apply Laplace transform to both sides of the differential equation.L{y′′+6y′+18y}=L{3δ(t−π)}L{y′′}+6L{y′}+18L{y}=3L{δ(t−π)}Using the properties of Laplace transform, we get: L{y′′} = s²Y(s) − s*y(0) − y′(0)L{y′} = sY(s) − y(0)where Y(s) is the Laplace transform of y(t).
Therefore,L{y′′+6y′+18y}=s²Y(s) − s*y(0) − y′(0) + 6(sY(s) − y(0)) + 18Y(s)Simplifying we get:Y(s)(s² + 6s + 18) - s - 1 = 3e^-πs
2: Apply the initial conditions to solve for constants.Using the initial condition, y(0) = 1, we get:Y(s)(s² + 6s + 18) - s - 1 = 3e^-πs ....(1)Using the initial condition, y′(0) = 6, we get:d/ds[Y(s)(s² + 6s + 18) - s - 1] s=0 = 6Y'(0) + Y(0) - 1Therefore,6(2)+1-1 = 12 ⇒ Y'(0) = 1
3: Convert the resulting expression back to the time domain.Solving equation (1) for Y(s), we get:Y(s) = 3e^-πs / (s² + 6s + 18) - s - 1Using partial fractions, we can write Y(s) as follows:Y(s) = A / (s + 3) + B / (s + 3)² + C / (s + 3)³ + D / (s - α) + E / (s - β)where α, β are roots of the quadratic s² + 6s + 18 = 0 with negative real parts, and A, B, C, D, E are constants we need to find
Multiplying through by the denominator of the right-hand side and solving for A, B, C, D, and E, we get:A = 3/2, B = -1/2, C = 1/6, D = 1/2, E = -1/2
Taking the inverse Laplace transform of Y(s), we get:y(t) = (3/2)e^-3t - (1/2)te^-3t + (1/6)t²e^-3t + (1/2)e^(-3+iπ)t - (1/2)e^(-3-iπ)twhere i is the imaginary unit.
Hence, the solution of the given symbolic initial value problem isy(t) = (3/2)e^-3t - (1/2)te^-3t + (1/6)t²e^-3t + (1/2)e^(-3+iπ)t - (1/2)e^(-3-iπ)t
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Which linear equality will not have a shared solution set with the graphed linear inequality? y > two-fifthsx 2 y < negative five-halvesx – 7 y > negative two-fifthsx – 5 y < five-halvesx 2
The linear equality that will not have a shared solution set with the graphed linear inequality is y > 2/5x + 2. So, option A is the correct answer.
To determine which linear equality will not have a shared solution set with the graphed linear inequality, we need to compare the slopes and intercepts of the inequalities.
The given graphed linear inequality is y > -5/2x - 3.
Let's analyze each option:
A. y > 2/5x + 2:
The slope of this inequality is 2/5, which is different from -5/2, the slope of the graphed inequality. Therefore, option A will not have a shared solution set.
B. y < -5/2x - 7:
The slope of this inequality is -5/2, which is the same as the slope of the graphed inequality. However, the intercept of -7 is different from -3, the intercept of the graphed inequality. Therefore, option B will have a shared solution set.
C. y > -2/5x - 5:
The slope of this inequality is -2/5, which is different from -5/2, the slope of the graphed inequality. Therefore, option C will not have a shared solution set.
D. y < 5/2x + 2:
The slope of this inequality is 5/2, which is different from -5/2, the slope of the graphed inequality. Therefore, option D will not have a shared solution set.
Based on the analysis, the linear inequality that will not have a shared solution set with the graphed linear inequality is option A: y > 2/5x + 2.
The question should be:
Which linear equality will not have a shared solution set with the graphed linear inequality?
graphed linear equation: y>-5/2x-3 (greater then or equal to)
A. y >2/5 x + 2
B. y <-5/2 x – 7
C. y >-2/5 x – 5
D. y <5/2 x + 2
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Answer:
b
Step-by-step explanation:
y<-5/2x - 7
. Determine the standard equation of the ellipse using the stated information.
Foci at (8,−1) and (−2,−1); length of the major axis is twelve units
The equation of the ellipse in standard form is _____.
b. Determine the standard equation of the ellipse using the stated information.
Vertices at (−5,12) and (−5,2); length of the minor axis is 8 units.
The standard form of the equation of this ellipse is _____.
c. Determine the standard equation of the ellipse using the stated information.
Center at (−4,1); vertex at (−4,10); focus at (−4,9)
The equation of the ellipse in standard form is ____.
a. The standard equation of the ellipse with foci at (8, -1) and (-2, -1), and a length of the major axis of 12 units is: ((x - 5)² / 6²) + ((y + 1)² / b²) = 1.
b. The standard equation of the ellipse with vertices at (-5, 12) and (-5, 2), and a length of the minor axis of 8 units is: ((x + 5)² / a²) + ((y - 7)² / 4²) = 1.
c. The standard equation of the ellipse with a center at (-4, 1), a vertex at (-4, 10), and a focus at (-4, 9) is: ((x + 4)² / b²) + ((y - 1)² / 9²) = 1.
a. To determine the standard equation of the ellipse with foci at (8, -1) and (-2, -1), and a length of the major axis of 12 units, we can start by finding the distance between the foci, which is equal to the length of the major axis.
Distance between the foci = 12 units
The distance between two points (x₁, y₁) and (x₂, y₂) is given by the formula:
√((x₂ - x₁)² + (y₂ - y₁)²)
Using this formula, we can calculate the distance between the foci:
√((8 - (-2))² + (-1 - (-1))²) = √(10²) = 10 units
Since the distance between the foci is equal to the length of the major axis, we can conclude that the major axis of the ellipse lies along the x-axis.
The center of the ellipse is the midpoint between the foci, which is (5, -1).
The equation of an ellipse with a center at (h, k), a major axis of length 2a along the x-axis, and a minor axis of length 2b along the y-axis is:
((x - h)² / a²) + ((y - k)² / b²) = 1
In this case, the center is (5, -1) and the major axis is 12 units, so a = 12/2 = 6.
Therefore, the equation of the ellipse in standard form is:
((x - 5)² / 6²) + ((y + 1)² / b²) = 1
b. To determine the standard equation of the ellipse with vertices at (-5, 12) and (-5, 2), and a length of the minor axis of 8 units, we can start by finding the distance between the vertices, which is equal to the length of the minor axis.
Distance between the vertices = 8 units
The distance between two points (x₁, y₁) and (x₂, y₂) is given by the formula:
√((x₂ - x₁)² + (y₂ - y₁)²)
Using this formula, we can calculate the distance between the vertices:
√((-5 - (-5))² + (12 - 2)²) = √(0² + 10²) = 10 units
Since the distance between the vertices is equal to the length of the minor axis, we can conclude that the minor axis of the ellipse lies along the y-axis.
The center of the ellipse is the midpoint between the vertices, which is (-5, 7).
The equation of an ellipse with a center at (h, k), a major axis of length 2a along the x-axis, and a minor axis of length 2b along the y-axis is:
((x - h)² / a²) + ((y - k)² / b²) = 1
In this case, the center is (-5, 7) and the minor axis is 8 units, so b = 8/2 = 4.
Therefore, the equation of the ellipse in standard form is:
((x + 5)² / a²) + ((y - 7)² / 4²) = 1
c. To determine the standard equation of the ellipse with a center at (-4, 1), a vertex at (-4, 10), and a focus at (-4, 9), we can observe that the major axis of the ellipse is vertical, along the y-axis.
The distance between the center and the vertex gives us the value of a, which is the distance from the center to either focus.
a = 10 - 1 = 9 units
The distance between the center and the focus gives us the value of c, which is the distance from the center to either focus.
c = 9 - 1 = 8 units
The equation of an ellipse with a center at (h, k), a major axis of length 2a along the y-axis, and a distance c from the center to either focus is:
((x - h)² / b²) + ((y - k)² / a²) = 1
In this case, the center is (-4, 1), so h = -4 and k = 1.
Therefore, the equation of the ellipse in standard form is:
((x + 4)² / b²) + ((y - 1)² / 9²) = 1
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Fill in the blank so that the resulting statement is true. The first step in solving ∣R+Ir=E for I is to obtain a single occurrence of I by............................I from the two terms on the left. The first step in solving IR+Ir=E for I is to obtain a single occurrence of I by.................................. I from the two terms on the left.
The first step in solving ∣R+Ir=E for I is to obtain a single occurrence of I by factoring out I from the two terms on the left. By using the distributive property of multiplication, we can rewrite the equation as I(R+r)=E.
Next, to isolate I, we need to divide both sides of the equation by (R+r).
This yields I=(E/(R+r)). Now, let's move on to the second equation, IR+Ir=E. Similarly, we can factor out I from the left side to get I(R+r)=E.
To obtain a single occurrence of I, we divide both sides by (R+r), resulting in I=(E/(R+r)).
Therefore, the first step in both equations is identical: obtaining a single occurrence of I by factoring it out from the two terms on the left and then dividing by the sum of R and r.
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Two tirequality experts examine stacks of tires and assign quality ratingsto each tire on a three-point scale. Let X denote the grade givenbe each expert A and Y denote the grade given by B. The followingtable gives the joint distribution for X and Y.
y
_F(x,y) 1 2 3___
1 0.10 0.05 0.02
x 2 0.10 0.35 0.05
3 0.03 0.10 0.20
Find μx and μy.
please show all steps to solve
The means μx and μy are 2.16 and 2.19, respectively.
To find the means μx and μy, we need to calculate the expected values for X and Y using the joint distribution.
The expected value of a discrete random variable is calculated as the sum of the product of each possible value and its corresponding probability. In this case, we have a joint distribution table, so we need to multiply each value of X and Y by their respective probabilities and sum them up.
The formula for calculating the expected value is:
E(X) = ∑ (x * P(X = x))
E(Y) = ∑ (y * P(Y = y))
Let's calculate μx:
E(X) = (1 * P(X = 1, Y = 1)) + (2 * P(X = 2, Y = 1)) + (3 * P(X = 3, Y = 1))
+ (1 * P(X = 1, Y = 2)) + (2 * P(X = 2, Y = 2)) + (3 * P(X = 3, Y = 2))
+ (1 * P(X = 1, Y = 3)) + (2 * P(X = 2, Y = 3)) + (3 * P(X = 3, Y = 3))
Substituting the values from the joint distribution table:
E(X) = (1 * 0.10) + (2 * 0.10) + (3 * 0.03)
+ (1 * 0.05) + (2 * 0.35) + (3 * 0.10)
+ (1 * 0.02) + (2 * 0.05) + (3 * 0.20)
Simplifying the expression:
E(X) = 0.10 + 0.20 + 0.09 + 0.05 + 0.70 + 0.30 + 0.02 + 0.10 + 0.60
= 2.16
Therefore, μx = E(X) = 2.16.
Now let's calculate μy:
E(Y) = (1 * P(X = 1, Y = 1)) + (2 * P(X = 1, Y = 2)) + (3 * P(X = 1, Y = 3))
+ (1 * P(X = 2, Y = 1)) + (2 * P(X = 2, Y = 2)) + (3 * P(X = 2, Y = 3))
+ (1 * P(X = 3, Y = 1)) + (2 * P(X = 3, Y = 2)) + (3 * P(X = 3, Y = 3))
Substituting the values from the joint distribution table:
E(Y) = (1 * 0.10) + (2 * 0.05) + (3 * 0.02)
+ (1 * 0.10) + (2 * 0.35) + (3 * 0.10)
+ (1 * 0.03) + (2 * 0.10) + (3 * 0.20)
Simplifying the expression:
E(Y) = 0.10 + 0.10 + 0.06 + 0.10 + 0.70 + 0.30 + 0.03 + 0.20 + 0.60
= 2.19
Therefore, μy = E(Y) = 2.19.
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