15.3 Exercise 11

Marvin calls on four customers. With probability $p_{1} = 0 . 6$ he makes a sale in each case. Geraldine calls on five customers, with probability $p_{2} = 0 . 5$ of a sale in each case. Customers who buy do so on an iid basis, and order an amount Y_i (in dollars) with common distribution:

Let D₁ be the total sales for Marvin and D₂ the total sales for Geraldine. Let $D = D_{1} + D_{2}$. Determine the distribution and mean and variance for D₁, D₂, and D. Determine $P \left(D_{1} \geq D_{2}\right)$ and $P \left(D \geq 1500\right)$, $P \left(D \geq 1000\right)$, and $P \left(D \geq 750\right)$.