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{\bf Question}

Suppose that a r.v. $X$ has the following probability mass
function (pmf):

$$f(x)=\left\{ \begin{array} {ll} cx, &amp; {\rm for}\ x=1,2,3,4,5;\\
0, &amp; {\rm otherwise} \end{array} \right.$$

Determine the value of the constant $c$.  Sketch the pmf of $X$
and find the following probabilities:

$$P\{X&lt;1\},\ P\{-1&lt;X&lt;3\},\ P\{X&gt;1\}$$

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{\bf Answer}

From ``the total probability is one",

$$\sum_{i=1}^5 ci = 1$$

and so $c=\frac{1}{15}$. Consequently

\hspace{1in}$P\{X=1\} = 0,$

\hspace{1in}$\displaystyle
P\{-1&lt;X&lt;3\}=P\{X=1\}+P\{X=2\}=\frac{1}{5}$

\hspace{1in}$\displaystyle P\{X&gt;1\}=1-P\{X=1\}=\frac{14}{15}$



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