Question

Use the second method of Problem$\mathbf{5}\mathbf{.}\mathbf{11}$to show that the expected number of successes in$\mathit{n}$ Bernoulli trials with probabilityof success is${\mathit{x}}^{\mathbf{-}}\mathbf{=}\mathit{n}\mathit{p}$. Hint: What is the expected number of successes in one trial?

Short answer

Expected number of successes in one trail, i.e.,$x^{-}=np$ .

Step-by-step solution

Given Information.

Assume that$x_1$ is first experiment,$x_2$ is second experiment and so on.

The equation$E\left(x+y\right)=E\left(x\right)+E\left(y\right)$ .

Definition of Bernoulli trails.

A Bernoulli trial is a random experiment with only two possible outcomes: "success" and "failure," with the probability of success being the same each time the experiment is run.

Find the expected number of successes in one trail

The expected number of successes:

$\begin{array}{rcl}{x}^{-}& =& E\left(x_1+x_2+...+x_n\right)\\ & =& E\left(x_1\right)+E\left(x_2\right)+...+E\left(x_n\right)\end{array}$

Substitute values,

$\begin{array}{rcl}{x}^{-}& =& p+p+p\\ & =& p\left(1+1+1\right)\\ & & np\end{array}$

Hence, expected number of successes in one trail, .