Unit 4: Statistics and Probability

Topic 4.14: Random variables: variance and continuous distributions (HL only) Questions

Practice 20 exam-style questions for IB Math AA SL Topic 4.14. Review the question stems below, then unlock the full Question Bank to access markschemes, model answers, and AI grading.

11 mark

2026

If all probabilities in a table sum to 0.9, then:

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2Write down1 mark

2026

A continuous random variable

X

has probability density function

f (x) = \frac{3 x ^{2}}{8}

for

0 \leq x \leq 2

, and

f (x) = 0

otherwise. It is given that

P (0.5 \leq X < 1.5) = \frac{13}{32}

.

The discrete random variable

Y

is defined as

X

rounded to the nearest integer, so

Y

takes the values 0, 1 or 2.

Write down

P (Y = 1)

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3Show that2 marks

2021

A continuous random variable

X

has probability density function

f (x) = 2 (1 - x)

for

0 \leq x \leq 1

, and

f (x) = 0

elsewhere. Show that

E (X) = \frac{1}{3}

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4Interpret1 mark

2026

The error

E

seconds in a stopwatch reading is the reading minus the true time. It is modelled by a continuous random variable with mean

0.3

. Interpret this mean in context.

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5Interpret1 mark

2026

A pendulum bob swings back and forth. Its horizontal displacement

D

metres from the rest position has a probability density function that is symmetric about

d = 0

, and

E (D) = 0

.

Interpret the result

E (D) = 0

in the context of the motion of the bob.

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6Find3 marks

2026

A discrete random variable X has P(X=1)=0.1, P(X=2)=0.4, P(X=3)=0.3, P(X=4)=c. Find c and E(X).

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7Write down1 mark

2026

For a continuous random variable

X

P (1 \leq X \leq 2) = 0.35

. A random variable

Y

equals

1

when

1 \leq X \leq 2

and equals

0

otherwise. Write down

P (Y = 1)

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8Find2 marks

2026

A spinner pays out X = 1, 2 or 5 dollars with P(X=1)=0.6, P(X=2)=0.3, P(X=5)=0.1. Find the expected payout E(X).

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9State1 mark

2023

A continuous random variable

X

has probability density function

f (x) = \frac{1}{4 b}

for

b \leq x \leq 5 b

, where

b > 0

, and

f (x) = 0

elsewhere. State

E (X)

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101 mark

2026

For a continuous random variable, P(X = 3) equals:

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11Show that2 marks

2021

A continuous random variable

X

has probability density function

f (x) = \frac{3 x ^{2}}{a ^{3}}

for

0 \leq x \leq a

, and

f (x) = 0

elsewhere. Show that

E (X) = \frac{3 a}{4}

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12Write down1 mark

2024

A random variable

X

has

P (X = x) = \frac{1}{3} (\frac{2}{3})^{x - 1}

for

x = 1, 2, 3, \dots

Write down an expression for

E (X)

in sigma notation.

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13Show that2 marks

2026

The graph of a probability density function is a triangle. It rises from

0

x = 0

to a height

h

x = 3

, then falls to

0

x = 8

. Show that

h = 0.25

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14Find1 mark

2025

A continuous random variable

T

has probability density function

f (t) = \frac{3 t ^{2}}{64}

for

0 \leq t \leq 4

, and

f (t) = 0

elsewhere. Find

1 \int 3 f (t) d t

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15Write down1 mark

2026

A continuous random variable

X

has probability density function

f (x) = \frac{3 x ^{2}}{8}

for

0 \leq x \leq 2

, and

f (x) = 0

elsewhere. Write down

P (X = 1.5)

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16Interpret1 mark

2026

A buoy bobs up and down on waves. Its height

H

cm above the calm-water level at a random moment is modelled by a continuous random variable with

E (H) = 0

. Interpret

E (H) = 0

in context.

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17Find1 mark

2024

A random variable

X

has

E (X) = \frac{1}{p}

. Given that

E (X) = 3.2

, find

p

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181 mark

2026

For any random variable, Var(X) equals:

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191 mark

2026

To find the median of a continuous variable you solve:

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20Find1 mark

2024

A random variable

X

has

Var (X) = \frac{1 - p}{p ^{2}}

. Find

Var (X)

when

p = 0.25

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