IB Mathematics Analysis and Approaches Topic AHL 4.14—properties of Discrete and Continuous Random Variables Questionbank (HL)

IB Mathematics Analysis and Approaches Topic AHL 4.14—properties of Discrete and Continuous Random Variables Questionbank (HL) | RevisionDojo

Practice IB Mathematics Analysis and Approaches (AA) Topic AHL 4.14—properties of Discrete and Continuous Random Variables with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 4.14—properties of Discrete and Continuous Random Variables and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

HLPaper 1

A discrete random variable $X$ gives the number of passengers boarding a bus during a 12-minute interval, with probability distribution shown below:

$x$	0	1	2	3	4
$P(X=x)$	0.1	$p$	0.3	$q$	0.2

A voucher value $D$ (in dollars) is determined by the formula $D = 6 X + 2$ (so the voucher increases by $6$ for each extra passenger).

Using $E (X) = 2.0$ , find the values of $p$ and $q$ .

[3]

Calculate the value of $E (D)$ , in dollars.

[2]

Hence, sketch the probability distribution for $X$ .

[2]

Question 2

HLPaper 1

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 $Var (X)$ .

[2]

Question 3

HLPaper 2

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

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

Given that $E (Z^{2}) = 2$ .

Write down an equation for $E (Z^{2}) = 2$ in terms of $q$ .

[2]

Determine the value of $q$ .

[2]

Question 4

HLPaper 1

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

$f (x) = {\frac{1}{2} (x - 1), 0, 1 \leq x \leq 3 otherwise$

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

[3]

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

[3]

Question 5

HLPaper 2

A discrete random variable $X$ takes values in the set ${0, 1, 2, 3, 4}$ , with probability mass function $P (X = x) = c (x^{2} + 1)$ , where $c$ is a constant.

Show that $c = \frac{1}{35}$ .

[3]

Construct the probability distribution table for $X$ .

[2]

Find $E (X)$ .

[3]

Find $Var (X)$ .

[4]

A new random variable is defined by $Y = 3 X - 2$ .

Find:

(i) $E (Y)$ ;

(ii) $Var (Y)$ .

[4]

The procedure of observing $X$ is repeated $n$ times independently. Let $T$ denote the number of trials in which $X = 4$ .

Find the smallest value of $n$ such that $P (T \geq 1) > 0.95$ .

[5]

IB Mathematics Analysis and Approaches AHL 4.14—properties of Discrete and Continuous Random Variables Questionbank

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IB Mathematics Analysis and Approaches AHL 4.14—properties of Discrete and Continuous Random Variables Questionbank

IB Mathematics Analysis and Approaches (AA) Topic AHL 4.14—properties of Discrete and Continuous Random Variables Questionbank

Unlock all 30 questions with a free account