Q. 8.1 - A First Course in Probability

Question

The number of automobiles sold weekly at a certain dealership is a random variable with an expected value of$16$. Give an upper bound to the probability that

$\left(a\right)$ next week’s sales exceed$18$;

$\left(b\right)$ next week’s sales exceed$25$.

Step-by-Step Solution

Answer

$$\begin{gathered} \left(a\right)P\{X>18\}\le \frac{E\left[X\right]}{18}=\frac{8}{9}=.89 \\ \left(b\right) P\{X>25\}\le \frac{E\left[X\right]}{25}=\frac{16}{25}=.64 \end{gathered}$$

Step 1 Given Information.

The number of automobiles sold weekly at a certain dealership is a random variable with an expected value of$16$.

Step 2 Part (a) Explanation.

Let the random variable$X$ represent the number of automobiles sold (in a week). It is given that$E\left[X\right]=16$.

By Markov's inequality, we get:

$P\{X>18\}\le \frac{E\left[X\right]}{18}=\frac{16}{18}=\frac{8}{9}=.89$

Step 3 Part (b) Explanation.

$P\{X>25\}\le \frac{E\left[X\right]}{25}=\frac{16}{25}=\frac{16}{25}=.64$