Math 132A

Combinations of Random Variables

Sum of Random Variables

• \(X_b = {}\) the amount a random customer spends in the bakery section.
• \(X_c = {}\) the amount a random customer spends in the coffee section.
• \(X_s = {}\) the amount a random customer spends in the souvenir section.

\(X = X_b + X_c + X_s\).

Difference of Random Variables

• \(R = {}\) the daily revenue of a small bakery.
• \(C = {}\) the daily cost of the small bakery.

Daily profit \(P = R - C\)

Multiple of a Random Variable

\(P = {}\) the daily profit of the small bakery.

Tax = 15% of the profit.

Daily tax: \(T = .15 P\)

Linear Combination of Random Variables

Given random variables \(X\) and \(Y\), the linear combination of \(X\) and \(Y\) with coefficients \(a\) and \(b\) is the random variable

\[aX + bY.\]

\[E(aX + bY) = aE(X) + bE(y)\]

Variability of Linear Combinations

If \(X\) and \(Y\) are independent random variables, then

\[\operatorname{Var}(aX + bY) = a^2\operatorname{Var}(X) + b^2\operatorname{Var}(Y))\]

Example (Tax)

A company operates two small bakeries in two different locations with different tax codes.

• In location 1, the tax is 12% of profit. The daily profit at that location is modeled by a random variable \(P_1\) with mean $530 and standard deviation $60.

• In location 2, the tax is 18% of profit. The daily profit at that location is modeled by a random variable \(P_2\) with mean $850 and standard deviation $30.

Example (Profit)

• Daily revenue: \(R\) with \(\mu_R = \$2050\) and \(\sigma_R = \$152\).
• Daily cost: \(C\) with \(\mu_C = \$950\) and \(\sigma_C = \$200\).