Answers
p is diagonal
q is diagonal
Or divide it into two and find using the Pythagoras’ theorem
c^2=a^2+b^2
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Problem 3 (20 points) In this problem we aim at utilizing the kerenl trick in Ridge regression and propose its kernalized version. Recall the Ridge regression training objective function:
f(w)= ||Xw - y|| 2 ^ 2 + lambda||w|| 2 ^ 2
for lambda > 0
a) Show that for w to be a minimizer of f(w) we must have X^ top Xw + lambda*Iw =X^ top y where X in mathbb R ^ (nd) is the data matrix with n samples each with d features, and I is iden- tity matrix (please check lectures for more details). Show that the minimizer of f(w) is w=(X^ top X + lambda*I )^ -1 X^ top y. Justify that the matrix X^ top X + lambda*I is invertible, for lambda > 0 (Hint: use SVD decomposition of data matrix X= U*Sigma V^ top and show all the eigenvalues of X^ top X + lambda*I are larger than zero).
b) Rewrite X^ top Xw + lambda*Iw =X^ top u as w= 1/lambda (X^ top y-X^ top Xw) . Based on this, show that we can write w =X^ top alpha for some alpha in mathbb R ^ n , and give an expression for a.
c) Based on the fact that w =X^ top alpha. explain why we say w is "in the span of the data."
d) Show that alpha=( lambda*I +XX^ top )^ -1 y. Note that X X^ top is the nn Gram (kernel) matrix for the standard vector dot product. (Hint: Replace w by X ^ top alpha in the expression for a, and then solve for a.)
e) Give a kernelized expression for the Xw, the predicted values on the training points. (Hint: Replace w by X ^ top alpha and a by its expression in terms of the kernel matrix X X^ overline top )
f) Give an expression for the prediction w * ^ top x for a test sample æ, not in the training set, where w * is the optimal solution. The expression should only involve a via inner products training data samples x_{i}, i = 1 ,...,n.
g) Based on (f), propose a kernalized version of the Ridge regression.
Answers
To obtain the prediction [tex]w_*^Tx[/tex] for a test sample x, we need to substitute [tex]w = X^[/tex]T\alpha into
a) To find the minimizer of the objective function f(w), we need to differentiate it with respect to w, set it equal to zero, and solve for w.
First, we expand the norm term:
[tex]||Xw - y||^2 = (Xw - y)^T(Xw - y) = w^TX^TXw - 2y^TXw + y^Ty[/tex]
Taking the derivative of f(w) with respect to w and setting it equal to zero, we get:
[tex]2(X^TXw - X^Ty)[/tex] + 2\lambda w = 0
Rearranging the terms, we have:
[tex]X^TXw[/tex] + \lambda Iw [tex]= X^Ty[/tex]
which implies that:
[tex](X^TX[/tex] + \lambda I)w [tex]= X^Ty[/tex]
To obtain the minimizer, we need to solve for w, which gives us:
[tex]w = (X^TX + \lambda I)^{-1}X^Ty[/tex]
To justify that [tex]X^TX[/tex] + \lambda I is invertible for lambda > 0, we can use the SVD decomposition of X:
X = U\Sigma [tex]V^T[/tex]
where U and V are orthogonal matrices and \Sigma is a diagonal matrix with the singular values of X. Then, we have:
[tex]X^TX = V\Sigma^TU^TU\Sigma V^T = V\Sigma^T\Sigma V^T[/tex]
Since \Sigma[tex]^T[/tex]\Sigma is also a diagonal matrix with non-negative entries, adding lambda I to [tex]X^TX[/tex] ensures that all eigenvalues are strictly positive, and hence the matrix is invertible.
b) Substituting [tex]X^Tu[/tex] for w in [tex]X^TXw[/tex] + \lambda Iw [tex]= X^Ty[/tex], we get:
[tex]X^TX(X^Tu)[/tex] + \lambda [tex]IX^Tu = X^Ty[/tex]
Simplifying, we get:
[tex]X^T[/tex]([tex]X^TX[/tex] + \lambda I)u =[tex]X^Ty[/tex]
Thus, we have:
[tex]u = (X^TX + \lambda I)^{-1}X^Ty[/tex]
Substituting this into [tex]X^Tu[/tex], we get:
[tex]w = X^T(X^TX + \lambda I)^{-1}X^Ty[/tex]
c) Since [tex]w = X^T[/tex] \alpha and [tex]X^T[/tex] represents a linear combination of the columns of X, we can say that w is a linear combination of the columns of X, and hence is "in the span of the data."
d) Substituting [tex]w = X^T[/tex]\alpha into the expression for a, we get:
[tex]a = \frac{1}{\lambda}(X^Ty - X^TX(X^T\alpha))[/tex]
Multiplying both sides by \lambda and rearranging, we get:
[tex]X^TX[/tex]\alpha + \lambda\alpha [tex]= X^Ty[/tex]
This can be rewritten as:
(\lambda I [tex]+ X^TX)[/tex]\alpha [tex]= X^Ty[/tex]
To obtain the expression for \alpha, we can simply solve for \alpha:
[tex]\alpha = (\lambda I + X^TX)^{-1}X^Ty[/tex]
Note that X^TX is the Gram (kernel) matrix for the standard vector dot product.
e) Substituting [tex]w = X^[/tex]T\alpha into the expression for Xw, we get:
[tex]Xw = XX^T\alpha[/tex]
Using the kernelized form of \alpha, we have:
[tex]Xw = XX^T(\lambda I + XX^T)^{-1}y[/tex]
f) To obtain the prediction [tex]w_*^Tx[/tex] for a test sample x, we need to substitute [tex]w = X^[/tex]T\alpha into
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Coach Cowley is going to the store to buy some turkey for lunch at the deli. If th turkey costs $3. 25 per pound, what equation represents thetotal cost of turkey, y, for the amount of pounds, x? whats the equation?
Answers
The equation represents the total cost of turkey and amount is y= 3.25x.
We have,
The turkey costs $3. 25 per pound.
let x be the amount in pounds and y be the total cost in dollar.
Then, the relation between x and y
Total cost = 3.25 (amount in poind)
y = 3.25 x
Thus, the required equation is y= 3.25x.
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Grandfather and his grandson started drinking tea and asked the grandson to bring some candy out of the box. The box contained 2 candies with nuts, 4 candies with caramel, 3 candies with marzipan and 1 candy with licorice. As the grandson was still small and the box was high on the shelf, he did not see what kind of candy he was taking. Find the probability that 1) 4 candies taken from the box blindly have different tastes; 2) 2 candies have the same taste; 3) 6 candies include 2 candies with marzipan, 2 candies with nuts and 2 candies with caramel.
a) Write down all the events that are asked to be probable using the symbols provided.
b) Find all probabilities asked by the number of combinations. For each calculation, present a calculation formula and then calculate
asked probability. (Please provide details on conversions and calculations.)
Answers
a) Let A denote the event that 4 candies taken have different tastes, B denote the event that 2 candies have the same taste, and C denote the event that 6 candies include 2 candies with marzipan, 2 candies with nuts and 2 candies with caramel.
b) The probability of event A is 1/210
The probability of event B is 5/126
The probability of event C is 3/70
To find the probability of event A, we need to count the number of ways to choose 4 candies out of 10, where each candy has a different taste. Thus, the probability of event A is given by:
P(A) = (2/10) * (4/9) * (3/8) * (1/7) = 1/210To find the probability of event B, we need to count the number of ways to choose 2 candies of the same taste and 2 candies of different tastes out of 10. There are 4 choices for the taste of the 2 candies that are the same, and 6 choices for the taste of the other 2 candies. Thus, the probability of event B is given by:
P(B) = (4/10) * (6/9) * (5/8) * (3/7) = 5/126To find the probability of event C, we need to count the number of ways to choose 2 candies with marzipan, 2 candies with nuts, and 2 candies with caramel out of 10. There are (3 choose 2) = 3 ways to choose 2 candies with marzipan, (2 choose 2) = 1 way to choose 2 candies with nuts, and (4 choose 2) = 6 ways to choose 2 candies with caramel. Thus, the probability of event C is given by:
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Give a general description of the steps used to determine the quadrant(s) in which the solutions lie for an angle in the range of 0 < θ < 2π (or 0 to 360 degrees) using terms such as inverse, reference angle, quadrants, etc.
Answers
To determine the quadrant(s) in which the solutions lie for an angle in the range of 0 < θ < 2π (or 0 to 360 degrees), there are several steps to follow.
Firstly, we need to identify the reference angle. This is the angle formed between the terminal arm of the angle and the x-axis in the standard position.
Next, we need to determine the sign of the angle, which is based on whether the terminal arm is located in the positive or negative x-axis, and the positive or negative y-axis.
Then, we need to use the inverse trigonometric functions (such as sin^-1, cos^-1, or tan^-1) to determine the exact angle measure. This step is important because it ensures that we obtain the angle measure within the desired range of 0 < θ < 2π.
Once we have the exact angle measure, we can determine the quadrant(s) in which the solution lies. This is based on the signs of the trigonometric functions in each quadrant. For example, if the sine and cosine are positive, the angle lies in the first quadrant. If the sine is positive and the cosine is negative, the angle lies in the second quadrant. If the sine and cosine are negative, the angle lies in the third quadrant. And if the sine is negative and the cosine is positive, the angle lies in the fourth quadrant.
In summary, to determine the quadrant(s) in which the solutions lie for an angle in the range of 0 < θ < 2π, we need to identify the reference angle, determine the sign of the angle, use the inverse trigonometric functions to find the exact angle measure, and then use the signs of the trigonometric functions in each quadrant to determine the quadrant(s) in which the solution lies.
A general description of the steps used to determine the quadrant(s) in which the solutions lie for an angle in the range of 0 < θ < 2π (or 0 to 360 degrees) involves understanding the angle, reference angle, and quadrant relationships. Here are the steps:
1. Convert the angle (θ) into standard position, which means placing the vertex at the origin and the initial side along the positive x-axis. If the angle is given in degrees, convert it to radians (if needed) using the conversion factor: 1 radian = 180/π degrees.
2. Identify the reference angle (α). The reference angle is the acute angle formed between the terminal side of the angle (θ) and the x-axis. To find the reference angle, use the following rules:
- If θ is in the first quadrant, α = θ
- If θ is in the second quadrant, α = π - θ
- If θ is in the third quadrant, α = θ - π
- If θ is in the fourth quadrant, α = 2π - θ
3. Determine the quadrant(s) in which the angle (θ) lies using the reference angle (α) and the inverse trigonometric functions.
The inverse trigonometric functions (e.g., sin⁻¹, cos⁻¹, and tan⁻¹) can help in finding the corresponding angle(s) for a given trigonometric function value. Depending on the function and value, one or two quadrants may be determined as solutions.
4. Once the quadrant(s) are identified, the solutions for the angle (θ) can be written using the reference angle (α) and the relevant inverse trigonometric function.
By following these steps, you can effectively determine the quadrant(s) in which the solutions lie for an angle within the specified range.
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To pay for the trailer, the company took out a loan that requires Amazon Rafting to pay the bank a special payment of $8,700 in 5 years and also pay the bank regular payments of $4,100 each year forever. The interėst rate on the loan is 14. 3 percent per year and the first $4,100 yearly payment will be paid in one year from today. What was the price of the trailer?
Answers
The price of the trailer was $74,041.54.
Let's start by finding the present value of the perpetual annuity payments of $4,100 per year, using the formula:
PV = PMT / r
where:
PV is the present value
PMT is the payment per period
r is the interest rate per period
Since the payments are made annually and the interest rate is 14.3% per year, the interest rate per period is also 14.3%. Thus:
PV = $4,100 / 0.143 = $28,671.33
At a current interest rate of 14.3%, Amazon Rafting would need to invest this much in order to receive permanent $4,100 yearly payments.
Now, using the following calculation, we can determine the price of the caravan using the present value of the perpetuity and the future value of the special payment:
[tex]FV = PV * (1 + r)^n + SP[/tex]
where:
FV is the future value
PV is the present value of the perpetuity
r is the interest rate per period (14.3% per year)
n is the number of periods (5 years)
SP is the special payment of $8,700 over 5 years
Substituting the values we have:
[tex]FV = $28,671.33 * (1 + 0.143)^5 + $8,700[/tex]
FV = $28,671.33 * 1.8333 + $8,700
FV = $74,041.54
Therefore, the price of the trailer was $74,041.54.
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Shannon found a stack of 100 collectible fantasy horse cards in her desk that she had forgotten about. She randomly looked at 20 cards and got 8 centaur, 7 pegasus, and 5 unicorn cards.
Based on the data, estimate how many unicorn cards are in the stack.
Answers
Shannon found a stack of 100 collectible fantasy horse cards in her desk that she had forgotten about. She randomly looked at 20 cards and got 8 centaur, 7 pegasus, and 5 unicorn cards.
Based on the data, estimate how many unicorn cards are in the stack.
If Shannon looked at 20 cards and got 5 unicorn cards, we can set up a proportion:
5/20 = x/100
Solving for x, we get:
x = 25
So, based on the data Shannon collected, we can estimate that there are around 25 unicorn cards in the stack.
Correctly use the wolframalpha method introduced in the Section 7.1 Learning Guidance and Section 7.1 Homework solutions (including your own correct using of parenthesis in the wolframalpha command), match X-Y the function z = x-y/1+x^2+y^2 given by Problem 30 on Page 392 with a graph and a contour map on Page 393. - Graph C, contour map II. - Graph C, contour map I. - Graph D, contour map I. - Graph D, contour map II.
Answers
To correctly use the wolframalpha command to match the function z = x-y/1+x^2+y^2 given by Problem 30 on Page 392 with a graph and a contour map on Page 393, you can follow the steps below:
1. Go to the wolframalpha website.
2. In the search bar, type "plot z = x-y/(1+x^2+y^2)" and hit enter.
3. The website will generate a 3D graph of the function.
4. To match the graph C and contour map II, click on the "More" button below the graph and select "Contour plot."
5. In the new window, select the second option from the left, which is the contour map.
6. Adjust the settings as necessary to match the colors and levels of the contour map on Page 393.
7. To match the graph C and contour map I, follow the same steps as above, but select the first option for the contour map.
8. To match the graph D and contour map I, click on the "More" button below the graph and select "Contour plot."
9. In the new window, select the first option from the left, which is the contour map.
10. Adjust the settings as necessary to match the colors and levels of the contour map on Page 393.
11. To match the graph D and contour map II, follow the same steps as above, but select the second option for the contour map.
It's important to correctly use parentheses in the wolframalpha command to ensure that the website understands the order of operations. In this case, we want to divide y by the sum of 1, x^2, and y^2 before subtracting it from x. Therefore, we need to enclose the denominator in parentheses, like this:
plot z = x-(y/(1+x^2+y^2))
By following these steps and using the correct wolframalpha command, you can match the function with the appropriate graph and contour map.
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A local repair shop charges $109 per hour to fix cars. A new water pump costs $249. 98. How many hours does the mechanic work, if the total cost of installing a new water pump is $849. 48?
Answers
The mechanic worked for 5.5 hours to install the new water pump.
Let's assume the mechanic worked for "x" hours to install the water pump.
The cost of the repair work will include the cost of the water pump as well as the labor charges.
The cost of the water pump is given as $249.98.
The labor charges can be calculated by multiplying the hourly rate by the number of hours worked.
So, the labor charges will be $109 multiplied by "x" hours, which is $109x.
Adding the cost of the water pump to the labor charges will give us the total cost of the repair work.
Therefore, we can write the equation as:
$849.48 = $109x + $249.98
Simplifying the equation, we get:
$599.50 = $109x
Dividing both sides by $109, we get:
x = 5.5
Therefore, the mechanic worked for 5.5 hours to install the new water pump.
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Let x,y ER" with y non-zero. The orthogonal projection of x onto the line determined by y is the vector Select one: O True O False of is the orthogonal projection of x onto the line determined by non-zero y then x - is orthogonal to y. Select one: O True False For unit vector y € R" the projection matrix for orthogonal projection onto the line determined by y is yy? Select one: True False
Answers
True. The orthogonal projection of vector x onto the line determined by non-zero vector y is indeed a vector. The projection creates a new vector that lies on the line determined by y and is orthogonal (perpendicular) to y.
2. If the orthogonal projection of x onto the line determined by non-zero y is the vector, then x - is orthogonal to y: True
If the orthogonal projection of x onto the line determined by non-zero y is the vector, then the difference between x and the projection (x - projection) will be orthogonal to y. This is because the projection is the closest point on the line determined by y to x, and thus the difference vector will be perpendicular to the line.
3. For unit vector y ∈ R, the projection matrix for orthogonal projection onto the line determined by y is yy^T: True
For a unit vector y, the projection matrix P for the orthogonal projection onto the line determined by y is given by P = yy^T, where y^T is the transpose of the vector y. This matrix is used to calculate the orthogonal projection of a vector onto the line determined by y.
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Vectors u and v are shown on the graph.
Answers
PART A: The component form of vector is: u = <4, 8> and v = <4, 7>
PART B: u + v = <8, 15>
PART C: 5u - 2v = <12, 26>
How to write vectors in component form?The component form of a vector is <x, y>.
PART A:
Looking at the graph, vector u is the displacement from (2, -6) to (6, 2). Thus,
u = (6, 2) - (2, -6)
u = (6-2, 2-(-6))
u = (4, 8)
In component form, u = <4, 8>
Vector v is the displacement from (7, 8) to (11, 1). Thus,
v = (11, 1) - (7, 8)
v = (11-7, 8-1)
v = (4, 7)
In component form, v = <4, 7>
PART B:
u + v = <4, 8> + <4, 7>
u + v = <4+4, 8+7>
u + v = <8, 15>
PART C:
5u - 2v = 5*<4, 8> - 2*<4, 7>
5u - 2v = <20, 40> - <8, 14>
5u - 2v = <20-8, 40-14>
5u - 2v = <12, 26>
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Which maps AABC to a triangle that is similar, but not congruent, to AABC?
A. reflection across the x-axis
B.
rotation 270° counterclockwise about the origin
C. translation right 2 units and up 3 units
D. dilation with scale factor 2 about the origin
Answers
The value of correct option for maps ΔABC to a triangle that is similar, but not congruent, to ΔABC are,
⇒ dilation with scale factor 2 about the origin
We have to given that;
To find correct option for maps ΔABC to a triangle that is similar, but not congruent, to ΔABC
Since, We know that;
For any translation the condition of congruency is not change.
But for any type of dilation condition of congruency for triangles are change.
Thus, The value of correct option for maps ΔABC to a triangle that is similar, but not congruent, to ΔABC are,
⇒ dilation with scale factor 2 about the origin
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can someone pls help.
Answers
Given: The solution to [tex]x^3[/tex] = [tex]-2-i[/tex] In polar form Is:
[tex]2 < 75^o, 2 < 195^o, 2 < 315^o[/tex]
Answer:
[tex]\large \boxed{\mathrm{ion \ even \ no \ fr}}[/tex]
Step-by-step explanation:
DO IT YOUR SELF [tex]\large \boxed{\mathrm{BOZO}}[/tex]
If our alternative hypothesis is mu > 1.2, and alpha is .05, where would qur critical region be? O In the upper 5% of the alternative distribution O In the lower 5% of the alternative distribution O In the lower and upper 2.5% of the null distribution O In the lower 5% of the null distribution O In the lower and upper 2.5% of the alternative distribution O In the upper 5% of the null distribution
Answers
The critical region would be in the tail of the null distribution corresponding to the alpha level (0.05), which is the upper 5%.
We have,
The critical region would be in the upper 5% of the null distribution.
This is because alpha is the probability of making a type I error (rejecting the null hypothesis when it is actually true),
In this case,
We are looking for evidence that the population mean is greater than 1.2.
Therefore,
The critical region would be in the tail of the null distribution corresponding to the alpha level (0.05), which is the upper 5%.
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The average mark on a chemistry test was 72% with a standard deviation of 8%. If sheila’s test had a z-score of 2. 2, what was her test score?
Answers
If Sheila’s test had a z-score of 2.2 then her test score was 89.6%.
We can use the formula for calculating the z-score of a value,
z = (x - μ) / σ, value we want to convert to a z-score is x, mean of the distribution is μ, standard deviation of the distribution is σ and z score is z. In this case, we know that the average mark on the test was 72%, which means μ = 72. We also know that the standard deviation was 8%, which means σ = 8. We know that Sheila's z-score was 2.2,
We can rearrange the formula to solve for x,
x = μ + zσ
Substituting in the values we know,
x = 72 + 2.2 * 8
x = 72 + 17.6
x = 89.6
Therefore, Sheila's test score was 89.6%.
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factory produces cylindrical bar: The production process can be modelling by normal distribution with mean length of Cm and a standard deviation of 0.25 CM, (a) What is the probability that a randomly selected bar has length shorter than 11.75 cm? 100 cylindrical bars are randomly selected for quality checking: (b) What are the mean and standard deviation of the sample mean length? (c) What is the probability that the sample mean length will be between 10.99 cm and [[.01 cm? (d) If 92.65% of the sample means are more than a specific length L, find L
Answers
We need to solve the equation 1 - Φ((L - Cm)/0.025) = 0.9265 for L. This can be done using a standard normal table or calculator.
(a) Let X be the length of the cylindrical bar. Then X ~ N(Cm, 0.25^2). We need to find P(X < 11.75).
Z = (X - Cm)/0.25 follows standard normal distribution.
P(X < 11.75) = P((X-Cm)/0.25 < (11.75-Cm)/0.25) = P(Z < (11.75-Cm)/0.25)
Using a standard normal table or calculator, we get P(Z < (11.75-Cm)/0.25) = Φ((11.75-Cm)/0.25)
where Φ is the cumulative distribution function of the standard normal distribution.
(b) The sample mean length, X, follows normal distribution with mean Cm and standard deviation σ/√n, where n = 100 is the sample size. So, X ~ N(Cm, 0.25/√100) = N(Cm, 0.025). Therefore, the mean of the sample mean length is Cm and the standard deviation of the sample mean length is 0.025.
(c) We need to find P(10.99 < X < 11.01), where X is the sample mean length.
Z = (X - Cm)/(0.025) follows standard normal distribution.
P(10.99 < X < 11.01) = P((10.99 - Cm)/(0.025) < Z < (11.01 - Cm)/(0.025))
Using a standard normal table or calculator, we get P((10.99 - Cm)/(0.025) < Z < (11.01 - Cm)/(0.025)) = Φ((11.01 - Cm)/(0.025)) - Φ((10.99 - Cm)/(0.025))
(d) Let L be the length such that 92.65% of the sample means are more than L. This means we need to find the value of L such that P(X > L) = 0.9265.
Z = (X - Cm)/(0.025) follows standard normal distribution.
P(X > L) = P(Z > (L - Cm)/0.025) = 1 - Φ((L - Cm)/0.025)
Therefore, we need to solve the equation 1 - Φ((L - Cm)/0.025) = 0.9265 for L. This can be done using a standard normal table or calculator.
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x to the tenth power multiplied by x to the fifth power
Answers
Answer :x^15
Step-by-step explanation:
You would combine components so it would be x^10x^5 you would add 5+10 and then you would get your answer
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A windowpane is 15 inches by 8 inches. What is the distance between opposite corners of the windowpane?
Answers
3. Let X and Y be independent random variables, with X having a Poisson(2) distribution and Y having the distribution given by the probability mass function values 0 2 probabilities 0.2 0.5 0.3 () Find ELY (1) Let F be the cumulative distribution function of X+Y. Find Fly). (c) Find P(X=Y). (d) A student calculates E[XY'1 = E[X]E[Y) = (2)((0.2)02 + (0.5)1+ (0.3)2) = 3.4 Is this calculation correct? If so, explain why each step is valid. If not, what mistake is the student making?
Answers
a. E[Y] = (0)(0.2) + (2)(0.5) + (4)(0.3) = 1.8 is the expected value for Y.
b. The cumulative distribution function of X+Y is P(X+Y = k) = Σ P(X=i)P(Y=k-i).
c. P(X=Y) is 0.3654.
d. Calculation is not correct. 3.6 is the correct value of E[XY].
What is variable?In mathematics, a variable is defined as an alphabetic character that expresses a numerical value or number. A variable is used to represent an unknown quantity in algebraic equations.
(a) The expected value of Y can be calculated as E[Y] = (0)(0.2) + (2)(0.5) + (4)(0.3) = 1.8.
(b) To find the cumulative distribution function of X+Y, we first note that the sum of two independent random variables has a probability mass function given by the convolution of their respective probability mass functions. That is,
P(X+Y = k) = Σ P(X=i)P(Y=k-i)
where the sum is taken over all possible values of i such that both P(X=i) and P(Y=k-i) are nonzero. Using this formula, we can compute the cumulative distribution function of X+Y as:
F(x) = P(X+Y ≤ x) = Σ P(X+Y = k) for k ≤ x
= Σ Σ P(X=i)P(Y=k-i) for k ≤ x
= Σ P(X=i) Σ P(Y=k-i) for k ≤ x
= Σ P(X=i) [tex]F_Y[/tex](x-i)
where [tex]F_Y[/tex](x) is the cumulative distribution function of Y. Since X has a Poisson(2) distribution, we can compute the cumulative distribution function of X+Y as:
F(x) = Σ P(X=i) F_Y(x-i)
= Σ [tex]e^{(-2)} (2^i / i!) (0.2P(Y=x-i=0) + 0.5P(Y=x-i=2) + 0.3P(Y=x-i=4))[/tex]
where P(Y=x-i=k) is the probability mass function of Y.
(c) P(X=Y) can be calculated as:
P(X=Y) = Σ P(X=i, Y=i)
= Σ P(X=i)P(Y=i) (since X and Y are independent)
= Σ [tex]e^{(-2)} (2^i / i!) (0.2)(0) + (0.5)(e^(-2))(2^i / i!) + (0.3)(e^{(-2)})(2^i / i!)^2[/tex]
= [tex]e^{(-4)} (0 + 0.5(2e^2/2) + 0.3(4e^2/4))[/tex]
= 0.3654
(d) The student's calculation is not correct. To see why, let's first note that E[XY] can be computed as:
E[XY] = E[E[XY|X]] = E[XE[Y|X]]
where E[Y|X] is the conditional expected value of Y given X. Since X and Y are independent, we have E[Y|X] = E[Y] = 1.8. Therefore,
E[XY] = E[XE[Y|X]] = E[X(1.8)] = 2(1.8) = 3.6
So the correct value of E[XY] is 3.6, which is twice the value calculated by the student. The mistake the student made was in assuming that E[XY] is equal to the product of E[X] and E[Y]. This is only true if X and Y are uncorrelated, which is not the case here since X and Y are independent but not identically distributed.
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Molly is painting a model house and needs to know how much paint she will need. She knows the surface area of the prism is 216 square inches and the surface area of the pyramid is 84 square inches.
What is the area Molly needs to paint? Follow the steps to solve this problem.
1. Which surface is shared by the two solids? What are the dimensions of this surface? (2 points)
2. There is another surface that Molly does not need to paint, because it won’t show when she displays the model house. Describe that surface. (2 points)
3. To find the area Molly needs to paint, she should add the surface areas of both solids and subtract:
Circle the correct answer. (3 points)
4. Find the area Molly needs to paint. Show your work, and be sure to include units with your answer. (3 points)
Answers
The study of the new CPMP Mathematics methodology described in Exercise 7 also tested staudent's abilities to solve word problems. This table shows how the CPMP and traditional groups performed. What can you conclude?
Math program n Mean Standard deviation
CPMP 320 57. 4 32. 1
Traditional 273 53. 9 28. 5
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The CPMP mathematics methodology seems to be more effective on average at helping students solve word problems compared to the traditional method.
There is a greater variation in performance among students in the CPMP group. We can compare the CPMP and traditional math program groups' performance on solving word problems by looking at the mean and standard deviation.
1) Observe the mean scores:
- CPMP: 57.4
- Traditional: 53.9
The CPMP group has a higher mean score than the traditional group, indicating that students in the CPMP group performed better on average.
2) Observe the standard deviations:
- CPMP: 32.1
- Traditional: 28.5
The CPMP group has a higher standard deviation than the traditional group, meaning that the scores in the CPMP group are more spread out. This suggests there's a greater range of performance levels among students in the CPMP group.
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Part A: Create your own experiment with 5 or more possible outcomes. (2 points)
Part B: Create the sample space for your experiment in Part A. Explain how you determined the sample space. (2 points)
Answers
Experiment: "Favorite Cake Flavor"
Hypothesis: Different people have different favorite cakes flavors.
Now, Data collection enables us to identify the cakes flavors both popular and unpopular amongst users.
Furthermore, data analysis based on factors such as age, gender, and location allows for determining if disparities exist between consumer flavor preferences.
The findings of this study would be beneficial to cakes manufacturers and retailers, as they could enhance comprehension of preferred buyer choices and eventuate their products accordingly.
Conclusively, improved knowledge of preferred flavor profiles would facilitate refining marketing methodologies to attract a broader target audience.
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What is the ratio of the area of the triangle to the area of the rectangle?
Answers
Answer:
The area of the triangle is one-half the area of the rectangle. So the correct answer is C.
what is the surface area of 8yd by 3yd by 1 yd?
Answers
Answer:
The surface area is 70 yards
Step-by-step explanation:
The formula for surface area is (SA)=2lw+2lh+2hw. Meaning it would be 2 times (8 times 3 + 8 times 1 + 3 times 1) which equals 70 yards.
A fabric designer is mapping out a new design.Part of the pattern is formed by a repeating polygon.The inital polygon has verticies(-7,3),(-4,6),(-1,3) and (-4,0).The next polygon is a translation of the first along the vector (3,-3).Which is not a vertex of the image
A:(-4,6) B:(-4,0) C:(-1,-3) D:(-1,3)
Answers
The vertex that is not part of the image is:
A:(-4,6)
What are the translations to an image?The translations to an image are represented as follows:
Translation left a units: f(x + a).Translation right a units: f(x - a).Translation up a units: f(x) + a.Translation down a units: f(x) - a.The vector notation of a translation is given as follows:
{x ± a, y ± a}
We have the next polygon is a translation of the first along the vector (3,-3).
The rule applied to each vertex of the image is:
(x, y) → (x + 3, y - 3).
Now, We have the vertices are:
(-7,3),(-4,6),(-1,3) and (-4,0).
Applying the translation rule, the vertices of the image are as follows:
(-4, 0), (-1, 3), (2, 0), (-1, -3).
The lone coordinate without a vertex of the image is (-4,6)
The correct option is (A)
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IM GIVING 30 POINTS TO WHOEVER ANSWERS THIS!
Graph g(x)=−|x+3|−2.
Use the ray tool and select two points to graph each ray.
Answers
Answer:
Step-by-step explanation:
Why did many Americans consider César Chávez a hero in the '60s and '70s?
What is the value of H?
Answers
Q2 + Let S be the part of the hyperbolic paraboloid z = x2-y located between the cylinders x² + y2 = 1 and x2 + y2 = 25. Calculate the area of the surfaces
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Therefore, the area of the surface S is approximately 1.14 square units.
Here We can parametrize the hyperbolic-paraboloid surface S as follows:
r(u,v) = (u, v, [tex]u^2[/tex] - v)
Here u is restricted to the interval [−1, 1] and v is restricted to the interval [−5, 5].
The area of the surface, we need to compute the magnitude of the cross product of the partial derivatives of r with respect to u and v:
|ru x rv| = |(1, 0, 2u) x (0, 1, -1)| = |(2u, 1, 0)| = [tex]\sqrt{(4u^2 + 1)}[/tex]
Therefore, the area of the surface is given by the double integral:
A = ∬S dS = ∫[tex]-5^5 * -1^1 \sqrt{ (4u^2 + 1)}[/tex] du/dv
We can evaluate this integral by making the substitution w = [tex]2u^2 + 1,[/tex] ,which gives:
A = ∫[tex]1^2 *1/4 \sqrt{w} dw[/tex]
= [tex](2/3) * w^{({3/2)}} |1^2 *1/4[/tex]
= [tex](2/3)(2 * \sqrt{5} - 1)[/tex]
So the area of the surface S is approximately 1.14 square units.
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An electronic assembly consists of two subsystems A and B. The following probabilities are known. P(A fails) = 0.35, P( A and B fail) = 0.22 and P(B fails alone) = 0.3. Evaluate (i) P (A fails alone) (ii) P (A fails given that B has failed) [ (Explain the solution using Venn diagram)
Answers
The Venn diagram can be used to visualize the different probabilities and relationships between events, and Bayes' theorem can be used to calculate conditional probabilities.
Let's start by drawing a Venn diagram to represent the probabilities given :
_________________
/ \
/ \
/ A∩B \
/ \
/_________________________\
/ \
/ \
/ A\B \
/ \
/______________ ____________\
\ /
\ /
|
B
We know that:
P(A fails) = 0.35, which means P(A works) = 0.65
P(A and B fail) = 0.22
P(B fails alone) = 0.3, which means P(B works) = 0.7
To find (i) P(A fails alone), we need to subtract the probability of A and B failing together from the probability of A failing:
P(A fails alone) = P(A fails) - P(A and B fail) = 0.35 - 0.22 = 0.13
Therefore, the probability of A failing alone is 0.13.
To find (ii) P(A fails given that B has failed), we need to use Bayes' theorem:
P(A fails | B fails) = P(A∩B) / P(B fails)
We already know that P(A∩B) = 0.22 and P(B fails) = 0.3. So, we can substitute these values to get:
P(A fails | B fails) = 0.22 / 0.3 = 0.7333...
Therefore, the probability of A failing given that B has failed is approximately 0.7333.
In summary, the Venn diagram can be used to visualize the different probabilities and relationships between events, and Bayes' theorem can be used to calculate conditional probabilities.
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Which ordered pairs represent points on the graph of this equation? Select all that apply.
–5/6x=y+1/6
(-5,4)
(-7,5)
(0,2)
(6,7)
(-5,-6)
(1,-1)
Answers
Answer:
(1,-1)
Step-by-step explanation:
Substituting the given point into the equation
[tex]-5/6x=y+1/6\\-5/6(1) = -1 + 1/6\\-5/6 = -5/6[/tex]
0.3z=2(z–8.5)———————————————————————————————
Answers
Answer:
z = 10
Step-by-step explanation:
Lets explain this to you,
Step one: Solve with the parenthesis first (2 times z is 2z and 2 times -8.5 is
-17)
Step two: Subtract 2z from both sides (0.3z - 2z = - 1.7) (2z - 2z = 0)
Step three: Divide negative 1.7 on both sides (-1.7 = -17)
Step four: Simplify - 1.7 / 17 which will finally be 10
There is an easier way but im also in class so I'll do it later!!!!