 # Discrete Random Variable

A discrete random variable can only take on either a finite or at most a countably infinite set of discrete values such as $0, 1, 2, 3, …$. Discrete random variables are often used to model the number of times an event occurs in a given time period. The sum of all the probabilities of the experiment is equal to $1$.

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The discrete random variable $X$ has probability mass function given by

$\text{P}(X=x)=\left\{ \begin{matrix} & \frac{ax}{n(n+1)}\,\,\,\,\text{for}\,\,x=1,2,…,n, \\ & 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{otherwise,} \\ \end{matrix} \right.$

where $a$ is a constant.

(i)

Show that $a=2$.



(i) Show that $a=2$.



(ii)

Find $\text{E}(X)$ in terms of $n$. You may use the result that $\sum\limits_{r=1}^{n}{{{r}^{2}}=\frac{n}{6}(n+1)(2n+1)}$.



(ii) Find $\text{E}(X)$ in terms of $n$. You may use the result that $\sum\limits_{r=1}^{n}{{{r}^{2}}=\frac{n}{6}(n+1)(2n+1)}$.



Two players ${{P}_{1}}$ and ${{P}_{2}}$ play a game with $n+1$ tokens labelled $1$ to $n+1$. Each player randomly picks one token without replacement and the player who picks the token with the smaller number loses. The amount of money lost by the losing player, in dollars, will be the number on the winning token. For example, if ${{P}_{1}}$ and ${{P}_{2}}$ pick the tokens labelled $5$ and $3$ respectively, ${{P}_{2}}$ loses $\$\,5$to${{P}_{1}}$. (iii) Explain why the probability of${{P}_{2}}$losing a game is$0.5$.  (iii) Explain why the probability of${{P}_{2}}$losing a game is$0.5$.  (iv) Given that${{P}_{2}}$loses, find the probability${{P}_{2}}$loses$\$\,(m+1)$ in terms of $m$ and $n$, where $m$ is such that $1\le m\le n$.





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