Question

A large auto dealership keeps track of sales and leases agreements made during each hour of the day. Let $X$= the number of cars sold and $Y$= the number of cars leased during the first hour of business on a randomly selected Friday. Based on previous records, the probability distributions of $X$and $Y$are as follows:

Define $D=X-Y$.

The dealership’s manager receives a $500$bonus for each car sold and a$300$ bonus for each car leased. Find the mean and standard deviation of the difference in the manager’s bonus for cars sold and leased. Show your work.

Short answer

From the given information, the mean and standard deviation are$340$ and$509.09$ respectively

Step-by-step solution

Given Information

It is given in the question that,

$\mu \chi =1.1,\sigma \chi =0.943$

$\mu \gamma =0.7,\sigma \gamma =0.64$

Explanation

The mean and standard deviation of total bonus of manager can be calculated as:

$$\begin{gathered} {\mu }_B=500\left(1.1\right)-300\left(0.7\right) \\ =340 \end{gathered}$$

$$\begin{gathered} {\sigma }_B=\sqrt{500(0.943{)}^2+300(0.64{)}^2} \\ =509.09 \end{gathered}$$