A Level · Mathematics 9709 · Probability & Statistics 2

Continuous random variables: A Level Mathematics questions with answers

Free Continuous random variables questions for A Level Mathematics (9709), part of Probability & Statistics 2. Each question has the answer and the marking points an examiner would look for.

Generate a full Continuous Random Variables worksheet →All Probability & Statistics 2 questions

Continuous random variables practice questions

Question 1Structured · 8 marksA* challenge

The random variable X has probability density function f(x) = kx2 for 1 ≤ x ≤ 4, and 0 otherwise.

(a)

Show that k = 43.

[2]
(b)

Find E(X), giving your answer in exact form.

[3]
(c)

Find the median of X.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)k(1 − 14) = 1
∫ kx−2 dx = 1 M1
correct result shown A1
2
(b)43 ln 4
∫ x × kx2 dx = k ln x M1
limits applied M1
43 ln 4 A1
3
(c)85
∫1m f(x) dx = ½ M1
k(1 − 1m) = ½ A1
85 A1
3
Question 2Structured · 6 marks

The random variable X has probability density function f(x) = kx(2 − x) for 0 ≤ x ≤ 2, and 0 otherwise.

(a)

Find the value of k.

[2]
(b)

Find E(X) and explain your answer using symmetry.

[2]
(c)

Find P(X < 1).

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)34
∫ = 1 M1
34 A1
2
(b)1
symmetric about x = 1 B1
1 B1
2
(c)0.5
integrate from 0 to 1 M1
0.5 A1
2
Question 3Structured · 8 marksA* challenge

The random variable X has probability density function f(x) = kx2 for 1 ≤ x ≤ 2, and 0 otherwise.

(a)

Show that k = 2.

[2]
(b)

Find E(X), giving your answer in exact form.

[3]
(c)

Find the median of X.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)k(1 − 12) = 1
∫ kx−2 dx = 1 M1
correct result shown A1
2
(b)2 ln 2
∫ x × kx2 dx = k ln x M1
limits applied M1
2 ln 2 A1
3
(c)43
∫1m f(x) dx = ½ M1
k(1 − 1m) = ½ A1
43 A1
3
Question 4Structured · 9 marksA* challenge

The random variable X has probability density function f(x) = kx2 for 0 ≤ x ≤ 3, and f(x) = 0 otherwise.

(a)

Find the value of k.

[2]
(b)

Find E(X) and Var(X).

[4]
(c)

Find the median of X, giving your answer correct to 3 significant figures.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)19
∫ kx2 dx = 1 M1
19 A1
2
(b)94, 2780
∫ x f(x) dx M1
E(X) = 94 A1
∫ x2 f(x) dx − E(X)2 M1
Var(X) = 2780 A1
4
(c)2.38
∫0m f(x) dx = ½ M1
m3 = 272 A1
2.38 A1
3
Question 5Structured · 6 marks

The random variable X has probability density function f(x) = kx(4 − x) for 0 ≤ x ≤ 4, and 0 otherwise.

(a)

Find the value of k.

[2]
(b)

Find E(X) and explain your answer using symmetry.

[2]
(c)

Find P(X < 1).

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)332
∫ = 1 M1
332 A1
2
(b)2
symmetric about x = 2 B1
2 B1
2
(c)0.156
integrate from 0 to 1 M1
0.156 A1
2
Question 6Structured · 4 marks

The random variable X has probability density function f(x) = kx2 for 0 ≤ x ≤ 2, and 0 otherwise.

(a)

Find k.

[2]
(b)

Find E(X).

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)38
∫ f(x) dx = 1 M1
38 A1
2
(b)1.5
∫ x f(x) dx M1
1.5 A1
2

More Probability & Statistics 2 questions

The Poisson distributionLinear combinations of random variablesSampling & estimationHypothesis tests

Want a mentor, not just a paper?

Riyaz Shaikh has mentored IGCSE, IB and A Level students for 15+ years, one-on-one, online and in Mumbai. The hardest questions on this site are the ones he uses to find a student's weak points.

Meet Riyaz →

Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in