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Question 1

The distribution of a discrete random variable $X$ is given by:

$P(X=-1)=2p, \quad P(X=2)=3q, \quad P(X=4)=1-2p-3q$

Find $p$ and $q$ such that the game is fair ( $E[X]=0$ ) and all probabilities are valid.

[6]

Question 2

The random variable $X$ takes values $-2, -1, 0, 1, 2$ with probabilities $P(X = -2) = p, \quad P(X = -1) = 2p, \quad P(X = 0) = 1 - 6p, \quad P(X = 1) = 2p, \quad P(X = 2) = p.$ Find the range of values for $p$ for which this is a valid probability distribution.

[3]

Question 3

A discrete random variable $X$ has the following probability distribution:

\begin{array}{|c|c|c|c|} \hline x & -2 & 0 & 5 \\ \hline P(X=x) & p & 2q & 1 - p - 2q \\ \hline \end{array}

Find an expression for $q$ in terms of $p$ such that $E[X] = 1$ , and determine the range of possible values for $p$ .

[7]

Question 4

A discrete random variable $X$ takes values $1, 2, 3, 4$ with probabilities given by: $\text{P}(X=1)=p$ $\text{P}(X=2)=2p$ $\text{P}(X=3)=3p$ $\text{P}(X=4)=1-6p$

Find the value of $p$ such that $\text{E}[X]=3$ , ensuring that the probability distribution is valid.

[4]

Question 5

The random variable $X$ takes values $1$ with probability $p$ and $-1$ with probability $1 - p$ . Find the value of $p$ so that the game is fair (i.e., $E[X] = 0$ ).

[3]

Question 6

A discrete random variable $X$ takes values $0, 3$ , and $5$ with probabilities $p, q$ , and $1-p-q$ respectively.

Given that $\text{E}[X]=4$ , express $q$ in terms of $p$ and determine the range of possible values for $p$ .

[5]

Question 7

The random variable $X$ has the following probability distribution:

\begin{aligned} P(X=0) &= p \\ P(X=1) &= 2p \\ P(X=2) &= 1 - 3p \end{aligned}

Find the set of values of $p$ for which this is a valid probability distribution.

[4]

Question 8

The random variable $X$ takes the value $2$ with probability $p$ and $-3$ with probability $1 - p$ . Find $p$ such that the expected value is zero.

[3]

Question 9

A discrete random variable $X$ has the probability distribution shown in the following table:

\begin{array}{|c|c|c|c|c|} \hline x & -2 & 0 & 1 & 3 \\ \hline P(X=x) & p & q & 2p & 1-3p-q \\ \hline \end{array}

Find an expression for $q$ in terms of $p$ such that the game is fair ( $E[X]=0$ ), and determine the range of possible values for $p$ .

[6]

Question 10

The discrete random variable $X$ takes values $-2, 1$ , and $3$ with probabilities $p, q$ , and $1-p-q$ , respectively.

Find $q$ in terms of $p$ such that $E[X]=0$ , and determine the range of possible values for $p$ .

[5]

Question 11

Let the discrete random variable $X$ have the following probability distribution:

\begin{aligned} \text{P}(X = -1) &= p \\ \text{P}(X = 0) &= 2p \\ \text{P}(X = 2) &= 1 - 3p \end{aligned}

Find the range of possible values for $p$ such that this is a valid probability distribution.

[3]

Question Type 3: Finding certain values of parameters such that the probability table holds true Exercises