Chapter 5, Problem 1CQ

To determine

Explain why the distribution is not a probability distribution.

Explanation of Solution

The rules for a discrete probability distribution are,

• The sum of the probabilities must equal to 1. That is, ${\displaystyle \sum P\left(x\right)}=1$
• Each probability must be between 0 and 1, inclusive. That is, $0\le P\left(x\right)\le 1$

Here, the random variable X takes values $x=6,7,8,9,10$. Here, sum of the probabilities is,

$\begin{array}{c}{\displaystyle \sum P\left(x\right)}=0.32+0.11+0.19+0.28+0.03\\ =0.93\\ \ne 1\end{array}$

Thus, the first rule is not satisfied.

Also, $0\le P\left(x\right)\le 1$,for all values of x. Hence, the second rule is satisfied.

Since the first rule is not satisfied the given distribution is not a probability distribution.