# 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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##### N1977 P1 Q12

A test consists of five multiple choice questions to each of which three answers are given, only one of which is correct. For each correct answer a candidate gets $2$ marks, but loses $1$ mark for each incorrect answer. A particular candidate answers all question purely by guesswork. Draw up a table to show all the possible total marks from $-5$ to $10$ and the probability associated with each mark for this candidate. Use the table to show that the expected mark is zero, and find the variance of this distribution.

##### N1978 P2 Q12

A fair die is thrown once and random variable $X$ denotes the score obtained. Write down $\text{E}\left( X \right)$ and show that $\text{Var}\left( X \right)=\frac{35}{12}$. The die is thrown twice, and ${{X}_{1}}$and${{X}_{2}}$ obtained from each attempt.
Tabulate the probability distribution of $Y=\left| {{X}_{1}}-{{X}_{2}} \right|$, and find $E\left( Y \right)$.
The random variable $Z$ is defined by $Z={{X}_{1}}-{{X}_{2}}$. State with reasons whether or not

(i)

$E\left( {{Z}^{2}} \right)=E\left( {{Y}^{2}} \right)$.

(ii)

$\text{Var}\left( Z \right)=\text{Var}\left( Y \right)$.

##### 2018 DHS P2 Q10

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 $a=2$.

[2]

(i) Show that $a=2$.

[2]

(ii)

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

[3]

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

[3]

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$0.5$. [1] (iii) Explain why the probability of${{P}_{2}}$losing a game is$0.5$. [1] (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 $1\le m\le n$.

[3]

[3]