Question

Suppose X is a discrete random variable that takes on integer values between 1 and 10, with variance Var(X) = 6. Suppose that you define a new random variable Y by observing the output of X and adding 3 to that number. What is the variance of Y? Suppose then you define a new random variable Z by observing the output of X and multiplying that by -4. What is the variance of Z?

Answer 1

Solution:

Given: Variance of X is : Var(X) = 6

Part a) Define a new random variable Y by observing the output of X and adding 3 to that number.

Y = X + 3

Then variance of Y is:

Var(Y) = Var(X+3)

Var(Y) = Var(X) +Var(3)

Var(Y) = 6 + 0

( Since variance of Constant is 0)

Thus

Var(Y) = 6

Part b) Define a new random variable Z by observing the output of X and multiplying that by -4.

Z = (-4)X

Then find Variance of Z.

Var(Z) = Var( (-4) X )

Var(Z) = (-4)² * Var(X)

(since Var( aX) = a² * Var(X) )

Var(Z) = 16   * 6

Var(Z) = 96