Question

MY NOTES

A retail dealer sells three brands of automobiles. For brand A, her profit per sale, X is normally distributed with parameters

(μ₁, σ₁²);

for brand B her profit per sale Y is normally distributed with parameters

(μ₂, σ₂²);

for brand C, her profit per sale W is normally distributed with parameters

(μ₃, σ₃²).

For the year, one-quarter of the dealer's sales are of brand A, one-half of brand B, and the remaining one-quarter of brand C. If you are given data on profits for

n₁,

n₂,

and

n₃

sales of brands A, B, and C, respectively, the quantity

U = 0.25X + 0.5Y + 0.25W

will approximate to the true average profit per sale for the year. Find the mean and variance for U. Assume that X, Y, and W are independent.

E(U)=

V(U)=

Identify the probability distribution for U.

Answer 1

As we are given here that: X, Y and W are independent random variables, therefore the linear combination of these normally distributed variables would also be a normal variable. The mean and variance of U are computed here as:

E(U) = 0.25E(X) + 0.5E(Y) + 0.25E(W)

This is the required expected value of U here.

Var(U) = 0.25²Var(X) + 0.5²Var(Y) + 0.25²Var(W)

This is the required variance of U here.

The probability distribution for U here is given as:

This is the required probability distribution of U here.