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Question 1

For a discrete random variable $X$ , the probability distribution is defined as $\text{P}(X=0)=p$ , $\text{P}(X=1)=2p$ , $\text{P}(X=2)=3p$ , and $\text{P}(X=3)=4p$ , where $p$ is a constant.

1.

(a) Find the value of $p$ .

[2]

2.

(b) Find the expected value $\text{E}(X)$ .

[2]

Question 2

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

$x$	$-1$	$0$	$1$	$2$	$3$
$P(X=x)$	$0.1$	$p$	$2p$	$0.3$	$q$

Given that $q = 2p$ , find the value of $p$ and the value of $q$ .

[5]

Question 3

Show that for a geometric distribution $P(X=k)=(1-p)p^k$ for $k=0, 1, 2, \dots$ , where $0 < p < 1$ , the probabilities sum to 1 and derive $E(X) = \frac{p}{1-p}$ .

[5]

Question 4

A random variable $Y$ has $P(Y=k)=Ck$ for $k=1,2,3$ , where $C$ is a constant.

1.

Determine the value of $C$ .

[2]

2.

Find $\text{Var}(Y)$ .

[4]

Question 5

Given a discrete random variable $X$ with $P(X=0)=0.2$ , $P(X=1)=p$ , and $P(X=2)=0.5$ , find the value of $p$ .

[2]

Question 6

Consider a discrete random variable $X$ following a geometric distribution on $k = 0, 1, 2, \dots$ with probability mass function $\text{P}(X = k) = a p^k$ , where $0 < p < 1$ .

1.

Find $a$ in terms of $p$ .

[2]

2.

Determine the expected value $\text{E}(X)$ .

[3]

Question 7

Determine the value of a parameter $p$ in a discrete probability distribution given the probabilities of all outcomes in terms of $p$ .

For a discrete random variable $X$ , $P(X=1)=p$ , $P(X=2)=2p$ , $P(X=3)=3p$ , and $P(X=4)=4p$ . Determine the value of $p$ .

[3]

Question 8

The question requires knowledge of the binomial distribution formula $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$ and the ability to solve a polynomial equation numerically (typically using a graphing display calculator - GDC).

In a binomial model $X \sim B(5, p)$ , it is known that $P(X=2) = 0.3$ . Find the possible values of $p$ .

[3]

Question 9

Let $X$ be a discrete random variable such that $P(X=k) = xk$ for $k = 1, 2, 3, 4, 5$ .

Given that $P(X=1) = x$ , find the value of $x$ and calculate $P(X > 3)$ .

[4]

Question 10

A Poisson random variable $X$ has parameter $\lambda$ . Prove that $\sum_{k=0}^\infty P(X=k)=1$ using the series expansion of $e^\lambda$ .

[3]

Question 11

Let $X$ be a random variable with $P(X=0)=0.3$ , $P(X=1)=0.4$ , $P(X=2)=p$ , and $P(X=3)=0.1$ . Find $p$ .

[2]

Question 12

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

Given that $\text{E}(X) = 2$ , find the value of $p$ .

[3]

Question Type 2: Finding the expected value of outcomes of random variables Exercises