91Ó°ÊÓ

Let \(\mathrm{X}\) be the random variable defined as the number of dots observed on the upturned face of a fair die after a single toss. Find the expected value of \(\mathrm{X}\).

A brush salesman sells door-to-door. His products are short and long brushes. The profit on the long brush is \(\$ .30\) and on the short one it is \(\$ .10\). The chances of selling a long brush are one out of ten calls, and the chances of selling a short one are, two out of ten calls. The chances of no sales are seven out of ten calls. Find the expected profit per call.

Find the expected number of boys on a committee of 3 selected at random from 4 boys and 3 girls.

Let \(\mathrm{X}\) be a random variable denoting the hours of life in an electric light bulb. Suppose \(\mathrm{X}\) is distributed with density function \(\mathrm{f}(\mathrm{x})=[1 /(1,000)] \mathrm{e}^{-\mathrm{x} / 1000}\) for \(x>0\) Find the expected lifetime of such a bulb.

Find the theoretical variance of the random variable with the following probability distribution. \begin{tabular}{|c|c|} \hline \(\mathrm{x}\) & \(\operatorname{Pr}(\mathrm{X}=\mathrm{x})\) \\ \hline 0 & \(1 / 4\) \\ \hline 1 & \(1 / 2\) \\ \hline 2 & \(1 / 8\) \\ \hline 3 & \(1 / 8\) \\ \hline \end{tabular}

The probability that a certain baseball player will get a hit on any given time at bat is \((3 / 10)\). If he is at bat one hundred times during the next month, find the theoretical mean and variance of \(\mathrm{x}\), the number of hits. Assume that the binomial distribution is applicable.

Find the variance of the random variable \(\mathrm{Y}=\mathrm{aX}\) where a is a constant and \(\mathrm{X}\) has variance, \(\operatorname{Var} \mathrm{X}=\sigma^{2}\)