Chapter 7, Problem 91SE

A retail dealer sells three brands of automobiles. For brand A, her profit per sale, X is normally distributed with parameters $\left({\mu }_1,{\sigma }_1^2\right)$; for brand B her profit per sale Y is normally distributed with parameters $\left({\mu }_2,{\sigma }_2^2\right)$; for brand C, her profit per sale W is normally distributed with parameters $\left({\mu }_3.{\sigma }_3^2\right)$. For the year, two-fifths of the dealer’s sales are of brand A, one-fifth of brand B, and the remaining two-fifths 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 $U=.4\bar{X}+.2\bar{Y}+.4\bar{W}$ will approximate to the true average profit per sale for the year. Find the mean, variance, and probability density function for U. Assume that X, Y, and W are independent.