The random variable X is normally distributed with mean 5 and standard deviation 25. The random variable Y is defined by Y = 2 + 4X. What are the mean and the standard deviation of Y ?

Answer:

The correct answer is "100".

Explanation:

The given values are:

X (mean) = 5

X (standard deviation) = 25

Variance of X will be:

= [tex]25^{2}[/tex]

= [tex]625[/tex]

The solution of the part first is:

The given equation is :

[tex]Y=2+4X[/tex]

On putting the value of x in above equation we get ,

[tex]Y=2+4\times 5 \\Y=2+20\\Y=22[/tex]

So we get the mean of Y is 22

For finding the S.D of Y

[tex]Y = 2+4X\\[/tex]

So,

variance of Y= Var(2+4X)

As we know that the variance of constant is zero

So, variance (2) =0

⇒ [tex]Variance(aX) = a^2Variance(X)[/tex]

⇒ [tex]Var(4X) = 42Var(X) = 16Var(X)[/tex]

So,

⇒ [tex]Variance(Y) = Variance (2) + Variance(4X)[/tex]

We know the variance of constant is zero

So, var(2)=0

⇒ [tex]Variance(Y)= 16\times 625\\ Variance(Y)=10000[/tex]

Thus the standard deviation of Y is:

= [tex](10000)^{0.5}[/tex]

= [tex]100[/tex]

Connor is driving on a long road trip. He currently has 9 gallons of gas in his car. Each hour that he drives, his car uses up 2 gallons of gas. How much gas would be in the tank after driving for 3 hours? How much gas would be left after t hours?