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 $\tau =\chi +\gamma$

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 manager’s total bonus $B$. Show your work.

Short answer

From the given information, the mean and standard deviation are $760$and $509.09$respectively.

Step-by-step solution

Given Information

It is given in the question that,

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

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

The manager receives a bonus of $500$for each sold car and $300$for each car sold.

Explanation

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

$$\begin{gathered} {\mu }_B=500\left(1.1\right)+300\left(0.7\right) \\ =760 \\ {\sigma }_B=\sqrt{500(0.943{)}^2+300(0.64{)}^2} \\ =509.09 \end{gathered}$$