Chapter 6.3, Problem 85E

a.

To determine

To explain: The random variable X being a binomial variable.

Explanation of Solution

Given:

  $\begin{array}{l}\text{Number of trials}\left(n\right)=350\\ \text{Probabaility of success}\left(p\right)=0.9990\end{array}$

Formula used:

The formulas compute the mean, variance and standard deviation of X are:

  $E\left(X\right)=np$

  $V\left(X\right)=np\left(1-p\right)$

  $SD\left(X\right)=\sqrt{V\left(X\right)}$

Calculation:

Therefore,

  $\begin{array}{l}X\sim Binomial\left(350,0.999\right)\\ P\left(X=x\right)=\left(\begin{array}{l}350\\ x\end{array}\right){0.999}^x{\left(1-0.999\right)}^{350-x}\end{array}$

b.

To determine

To compute: The mean and standard deviation of X .

Explanation of Solution

Calculation:

The mean and standard deviation of X can be calculated as:

  $\begin{array}{c}E\left(X\right)=350\times 0.999\\ =349.65\end{array}$

  $\begin{array}{c}V\left(X\right)=350\times 0.999\left(1-0.999\right)\\ =0.3496\end{array}$

  $\begin{array}{c}SD\left(X\right)=\sqrt{0.3496}\\ =0.5913\end{array}$

Thus, the mean and standard deviation are 349.65 and 0.5913.

c.

To determine

To compute: The probability that X less than or equal to 348.

Answer to Problem 85E

The required probability is 0.0486.

Explanation of Solution

Calculation:

The probability that X less than or equal to 348 is calculated as:

  $\begin{array}{c}P\left(X\le 348\right)=1-P\left(X=349\right)-P\left(X=350\right)\\ =1-\left(\begin{array}{l}350\\ 349\end{array}\right){0.999}^{349}{\left(1-0.999\right)}^{350-349}-\left(\begin{array}{l}350\\ 350\end{array}\right){0.999}^{350}{\left(1-0.999\right)}^{350-350}\\ =1-0.24684-0.70456\\ =0.0486\end{array}$

The required probability is 0.0486.