CAIE A-Level Mathematics 6.3 Continuous Random Variables Question Bank

CAIE A-Level Mathematics 6.3 Continuous Random Variables Question Bank
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise integrating probability density functions to find constants, probabilities, expectations, medians and percentiles.

Exam points

  • set the total area under a density function equal to 1 to determine constants
  • integrate f(x) or xf(x) over the correct support for probabilities and means
  • find a median or percentile by equating the cumulative probability to its target

Covered syllabus points

6.3.1—Continuous random variables6.3.2—Density functions

Question 7

[Maximum number: 7]

The time, in minutes, taken by students to complete a test is modelled by the random variable X with probability density function

f(x)={−34(x−3)(x−5)3⩽x⩽50 otherwise f(x)= \begin{cases}-\frac{3}{4}(x-3)(x-5) & 3 \leqslant x \leqslant 5 \\ 0 & \text { otherwise }\end{cases}

Question 7(a)

(a)

Find the probability that a randomly chosen student takes longer than 4.5 minutes to complete the test.

[ 4 ]

Question 7(b)

(b)

Write down the median of X.

[ 1 ]

Question 7(c)

(c)

Without performing an integration, use your answer to part (a) to find P(3.5<X<4.5).

[ 2 ]

Question 6

[Maximum number: 11]
Figure for Question 6 — CAIE A-Level Mathematics

The diagram shows the graph of the probability density function, f, of a random variable X. The graph is a straight line from ( 0, a ) to ( b, 0 ) where a and b are constants. Elsewhere f(x)=0.

Question 6(a)

(a)

Find an expression for b in terms of a.

[ 2 ]

Question 6(b)

(b)

Given that E(X)=49\mathrm{E}(X)=\frac{4}{9} find the value of a.

[ 5 ]

Question 6(c)

(c)

（c）Using the value of a found in part（b）find the value of k such that P(X<k)=34\mathrm{P}(X<k)=\frac{3}{4} ．

[ 4 ]