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 probability density function

f(x)=\begin{cases} k\,x\,e^{-x/3},& x\ge 0,\\ 0,& x<0. \end{cases}

Determine the value of the constant $k$ .

[2]

Find $E(X)$ and $\mathrm{Var}(X)$ .

[3]

Show that the mode of $X$ is $3$ . Then, by solving $\displaystyle \int_{0}^{m} f(x)\,dx=\tfrac12$ , find the median $m$ correct to three significant figures (calculator required).

[3]

Let $Y=2X+5$ . Find $E(Y)$ and $\mathrm{Var}(Y)$ . Hence compute $P(10\le Y\le 20)$ .

[3]

Find $P(X>6\mid X>3)$ and give your answer in exact form.

[2]

Question 6

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 7

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 8

HLPaper 2

A box contains two types of biased coins. One coin is drawn at random and tossed $n=8$ times.

With probability $0.4$ , a Type A coin is drawn; for Type A, the probability of heads on any toss is $p_A=0.35$ .
With probability $0.6$ , a Type B coin is drawn; for Type B, the probability of heads on any toss is $p_B=0.65$ .

Let $X$ be the total number of heads in the 8 tosses. Assume tosses are independent conditional on the coin type.

Write down the conditional distributions of $X\mid A$ and $X\mid B$ . Hence find $P(X=3)$ .

[4]

Find $E(X)$ and $\mathrm{Var}(X)$ without expanding the full mixture PMF, by using the laws of total expectation and total variance.

[5]

The experiment resulted in exactly $x=6$ heads. Compute $P(B\mid X=6)$ .

[3]

A game pays € $g(x)=2x-9$ . Decide whether to Play (before seeing any data).

[2]

After observing that the first two tosses are both heads, you may Play the same game on the remaining six tosses. Compute $E[g(X)\mid \text{first two are heads}]$ and decide.

[4]

Question 9

HLPaper 1

A continuous random variable $Y$ has the probability density function $g$ given by

g(y)=\left\{\begin{array}{cl} \frac{\pi y}{36} \sin \left(\frac{\pi y}{6}\right), & 0 \leqslant y \leqslant 6 \\ 0, & \text { otherwise } \end{array} .\right.

Find $P(0 ≤ Y ≤ 3)$ .

[7]

Question 10

HLPaper 1

Let $Y$ be a continuous random variable with probability density function $f(y) = \begin{cases} \frac{3}{8}y^2, & 0 \leq y \leq 2 \\ 0, & \text{otherwise} \end{cases}$ .

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

[3]

Find the expected value of $Y$ .

[4]

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