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Question

# Your company, Sonic Video, Inc., has conducted research that shows the following probability distribution, where X is the number of video arcades in a randomly chosen city with more than 500,000 inhabitants:$\begin{matrix} \text{x} & \text{0} & \text{1} & \text{2} & \text{3} & \text{4} & \text{5} & \text{6} & \text{7} & \text{8} & \text{9}\\ \text{P(X=x)} & \text{.07} & \text{.09} & \text{.35} & \text{.25} & \text{.15} & \text{.03} & \text{.02} & \text{.02} & \text{.01} & \text{.01}\\ \end{matrix}$a. Compute the mean, variance, and standard deviation (accurate to one decimal place). b. As CEO of Startrooper Video Unlimited, you wish to install a chain of video arcades in Sleepy City, U.S.A. The city council regulations require that the number of arcades be within the range shared by at least 75% of all cities. What is this range? What is the largest number of video arcades you should install so as to comply with this regulation?

Solution

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#### a.

For a finite random variable $X$ taking on values $x_{1},x_{2},\ldots,x_{n}$,

the variance of $X$ is

\begin{align*} \sigma^{2}& =E([X-\mu]^{2}) \\ &=(x_{1}-\mu)^{2}P(X=x_{1})+(x_{2}-\mu)^{2}P(X=x_{2})+\cdots+(x_{n}-\mu)^{2}P(X=x_{n}) \\ & =\displaystyle \sum_{i}(x_{i}-\mu)^{2}P(X=x_{i}) . \end{align*}

The standard deviation of $X$ is then the square root of the variance:

$\sigma=\sqrt{\sigma^{2}}.$

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