Question

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 1

The correct answer is "100".

Step-by-step explanation:

The given values are:

X (mean) = 5

X (standard deviation) = 25

Variance of X will be:

=
`25^(2)`

=
`625`

The solution of the part first is:

The given equation is :

`Y=2+4X`

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

`Y=2+4* 5 \\Y=2+20\\Y=22`

So we get the mean of Y is 22

For finding the S.D of Y

`Y = 2+4X\\`

So,

variance of Y= Var(2+4X)

As we know that the variance of constant is zero

So, variance (2) =0

⇒
`Variance(aX) = a^2Variance(X)`

⇒
`Var(4X) = 42Var(X) = 16Var(X)`

So,

⇒
`Variance(Y) = Variance (2) + Variance(4X)`

We know the variance of constant is zero

So, var(2)=0

⇒
`Variance(Y)= 16* 625\\ Variance(Y)=10000`

Thus the standard deviation of Y is:

=
`(10000)^(0.5)`

=
`100`