AHL 4.14—Properties of discrete and continuous random variables Questionbank

Practice AHL 4.14—Properties of discrete and continuous random variables with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 2

A discrete random variable $X$ represents the number of successful attempts in a sequence of 5 independent trials, each with a success probability of 0.3 . The probability distribution of $X$ follows a binomial distribution.

Write down the probability mass function of $X$ .

[2]

Calculate the expected value $E(X)$ .

[2]

Find the variance $\operatorname{Var}(X)$ .

[2]

Sketch the probability distribution of $X$ .

[3]

Question 2

HLPaper 2

A discrete random variable $Z$ represents the number of errors in a code block, with probability distribution:

$z$	0	1	2	3
$P(Z=z)$	0.4	0.3	$q$	0.1

Given that $E\left(Z^{2}\right)=1.5$ , find the value of $q$ .

Write down the equation for $E\left(Z^{2}\right)$ .

[2]

Determine the value of $q$ .

[2]

Question 3

HLPaper 2

A discrete random variable $W$ represents the number of calls received in an hour, with probability distribution:

$w$	0	1	2	3	4
$P(W=w)$	0.2	0.25	$p$	0.15	0.1

Given that $\operatorname{Var}(W)=1.6875$ , find the value of $p$ .

Calculate $E(W)$ .

[2]

Determine the value of $p$ .

[3]

Question 4

HLPaper 2

A continuous random variable $X$ has the probability density function:

f(x)= \begin{cases}\frac{2}{3} \sin \left(\frac{\pi x}{4}\right), & 0 \leq x \leq 4 \\ 0, & \text { otherwise }\end{cases}

Verify that $f(x)$ is a valid probability density function.

[3]

Find $P(1 \leq X \leq 3)$ .

[3]

Calculate the mode of $X$ .

[2]

Question 5

HLPaper 2

A continuous random variable $X$ has the probability density function:

f(x)= \begin{cases}k e^{-x}, & 0 \leq x \leq 2 \\ 0, & \text { otherwise }\end{cases}

Find the value of $k$ .

[3]

Find the cumulative distribution function $F(x)$ .

[3]

Calculate the interquartile range of $X$ .

[3]

Question 6

HLPaper 2

A discrete random variable $T$ represents the number of customers served in an hour, with the probability distribution:

$t$	0	1	2	3	4
$P(T=t)$	0.1	0.3	0.4	$p$	0.05

Given that $E(T)=1.95$ , find the value of $p$ .

Set up the equation for $E(T)$ .

[2]

Determine the value of $p$ .

[2]

Question 7

HLPaper 2

A continuous random variable $X$ has the probability density function defined by:

f(x)= \begin{cases}k x(2-x), & 0 \leq x \leq 2 \\ 0, & \text { otherwise }\end{cases}

where $k \in \mathbb{R}^{+}$ .

Find the value of $k$ .

[3]

Calculate $E(X)$ .

[3]

Find the median of $X$ .

[3]

Question 8

HLPaper 2

A continuous random variable $W$ has the probability density function:

f(w)= \begin{cases}\frac{3}{8}\left(w^{2}+1\right), & 0 \leq w \leq 2 \\ 0, & \text { otherwise }\end{cases}

Verify that $f(w)$ is a valid probability density function.

[3]

Find the cumulative distribution function $F(w)$ .

[3]

Sketch the graph of $f(w)$ .

[2]

Question 9

HLPaper 2

A discrete random variable $Z$ represents the number of items sold in a day, with probability distribution:

$z$	0	1	2	3	4
$P(Z=z)$	0.15	0.25	$p$	0.2	$q$

Given that $E(Z)=2.1$ and $\operatorname{Var}(Z)=1.69$ , find the values of $p$ and $q$ .

Set up the equations for $E(Z)$ and the sum of probabilities.

[2]

Calculate $E\left(Z^{2}\right)$ and set up the variance equation.

[2]

Solve for $p$ and $q$ .

[3]

Question 10

HLPaper 2

A continuous random variable $Y$ has the probability density function:

f(y)= \begin{cases}k y e^{-2 y}, & y \geq 0 \\ 0, & \text { otherwise }\end{cases}

Find the value of $k$ .

[3]

Calculate the median of $Y$ .

[3]

z

P(Z=z)

0.4

0.3

q

0.1

w

P(W=w)

0.2

0.25

p

0.15

0.1

t

P(T=t)

0.1

0.3

0.4

p

0.05

z

P(Z=z)

0.15

0.25

p

0.2

q