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# An office has four copying machines, and the random variable $X$ measures how many of them are in use at a particular moment in time. Suppose that $P(X = 0) = 0.08\,, \,P(X = 1) = 0.11\,,\, P(X = 2) = 0.27\,,\,$ and $P(X = 3) = 0.33$ . $\mathbf{(a)}$What is $P(X=4) ?$ $\mathbf{(b)}$ Draw a line graph of the probability mass function. $\mathbf{(c)}$ Construct and plot the cumulative distribution function.

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$\underline{\color{#4257b2}{\textbf{ Probability Mass Function ( Distribution ) }}}$

$\color{#4257b2}{ \text{Let } X \text{ be a discrete random variable and } \underline{P(X = x_i) = p_i}} .$

$\color{#4257b2}{ \text{The } \text {probability mass function (p.m.f.) } \text{of } X \text { is a set of probability values } pi \text{ , and} }$

$\color{#4257b2}{\text{these probability values must satisfy two conditions : }}$

\begin{align*} \color{#4257b2}{(1) \,\,\,0 \le p_i \le 1}\\ \\ \color{#4257b2}{(2)\,\,\, \sum_i p_i =1} \end{align*}

$\underline{\color{#4257b2}{\textbf{Cumulative Distribution Function }}}$

$\color{#4257b2}{ \text{The cumulative distribution function (c.d.f.) of } X \text{ is the function } }$

$\color{#4257b2}{\,\,\,F(x)=P(X \le x)}$

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