CAIE A-Level Mathematics 6.3.2 Probability Density Functions

CAIE A-Level Mathematics 6.3.2 Probability Density Functions
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise integrating a continuous density over its support to find constants and probabilities, then locating percentiles or calculating expectation and variance from weighted…

How this is tested

  • set the integral of f(x) over its full support equal to one to determine an unknown constant
  • integrate f(x) between stated bounds and use symmetry or complements for the required area
  • solve an area equation for a median or percentile and use x f(x) or x²f(x) for moments

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 ]