Chapter 6.1, Problem 5E

a.

To determine

To show: The probability distribution is a legitimate probability distribution.

Explanation of Solution

Given:

The probability distribution is:

X	1	2	3	4	5	6	7	8	9
Probability	0.301	0.176	0.125	0.098	0.079	0.067	0.058	0.051	0.046

Calculation:

The sum of probabilities is:

  $\begin{array}{c}\text{Sum of probabilities = 0}\text{.301+0}\text{.176+}\mathrm{....}\text{+0}\text{.046}\\ \text{=1}\end{array}$

The probabilities are between 0 and 1 and the addition of probabilities is equal to 1. Thus, the provided probability distribution is a legitimate probability distribution.

b.

To determine

To construct: The graph for the probability distribution and interpret it.

Explanation of Solution

Graph

The graph could be constructed as:

  

Interpretation:

The most of the data is on the right side of the graph. Thus, the distribution is skewed to the right.

c.

To determine

To describe: The $X\ge 6$ . Also, compute the $P\left(X\ge 6\right)$

Answer to Problem 5E

The probability is 0.222.

Explanation of Solution

Calculation:

  $X\ge 6$ implies that 6 or more digits. In simple words, it means at least 6 digits.

  $P\left(X\ge 6\right)$ can be calculated as:

  $\begin{array}{c}P\left(X\ge 6\right)=P\left(X=6\right)+P\left(X=7\right)+P\left(X=8\right)+P\left(X=9\right)\\ =0.067+0.058+0.051+0.046\\ =0.222\end{array}$

Thus, the required probability is 0.222.

d.

To determine

To express: The event “first digit is at most 5” in terms of X and compute it.

Answer to Problem 5E

The probability is 0.778.

Explanation of Solution

Calculation:

The event “first digit is at most 5” in terms of X could be written as $X\le 5$ .

  $P\left(X\le 5\right)$ can be calculated as:

  $\begin{array}{c}P\left(X\le 5\right)=P\left(X=1\right)+P\left(X=2\right)+\mathrm{.....}+P\left(X=5\right)\\ =0.301+0.176+\mathrm{....}+0.079\\ =0.778\end{array}$

Thus, the required probability is 0.222.