IB · Mathematics: Analysis & Approaches SL & HL · Statistics & probability

Continuous random variables: IB Mathematics: Analysis & Approaches questions with answers

Free Continuous random variables questions for IB Mathematics: Analysis & Approaches (SL & HL), HL only, part of Statistics & probability. Each question has the answer and the marking points an examiner would look for.

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Continuous random variables practice questions

Question 1Structured · 7 marks

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

(a)

Find the value of k.

[2]
(b)

Find E(X).

[2]
(c)

Find the median of X.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)364
k[x33]04 = 1 M1
k = 364 A1
2
(b)3
∫04x × 364x2dx (M1)
3 A1
2
(c)4∛2
∫0m364x2dx = 12 M1
m3 = 642 A1
m = 4∛2 A1
3
Question 2Structured · 8 marks

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

(a)

Find the value of k.

[3]
(b)

Write down E(X), giving a reason.

[2]
(c)

Find P(X < 1).

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)332
∫04kx(4 − x)dx = 1 M1
k[4x22 − x33]04 = 323k A1
k = 332 A1
3
(b)2
the pdf is symmetric about x = 2 R1
E(X) = 2 A1
2
(c)532
∫01332x(4 − x)dx (M1)
correct integration and substitution (A1)
532 A1
3
Question 3Structured · 7 marks

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

(a)

Find k.

[2]
(b)

Find E(X).

[2]
(c)

Find the median of X.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)19
∫ f(x) dx = 1 (M1)
19 A1
2
(b)94
∫ x f(x) dx (M1)
94 A1
2
(c)2.38
∫ from 0 to m = 0.5 (M1)
m3 = 272 (A1)
2.38 A1
3
Question 4Structured · 7 marks

A continuous random variable X has probability density function f(x) = kxe−x for 0 ≤ x ≤ 3, and f(x) = 0 otherwise.

(a)

Find the value of k.

[2]
(b)

Find E(X).

[2]
(c)

Find the median of X.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)1.25
∫03kxe−xdx = 1 (M1)
k = 1.25 A1
2
(b)1.44
∫03x·kxe−xdx (M1)
1.44 A1
2
(c)1.38
∫0mkxe−xdx = 0.5 (M1)
attempt to solve using GDC (M1)
m = 1.38 A1
3
Question 5Structured · 8 marks

A continuous random variable X has probability density function f(x) = kx(6 − x) for 0 ≤ x ≤ 6, and f(x) = 0 otherwise.

(a)

Find the value of k.

[3]
(b)

Write down E(X), giving a reason.

[2]
(c)

Find P(X < 32).

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)136
∫06kx(6 − x)dx = 1 M1
k[6x22 − x33]06 = 36k A1
k = 136 A1
3
(b)3
the pdf is symmetric about x = 3 R1
E(X) = 3 A1
2
(c)532
∫03/2136x(6 − x)dx (M1)
correct integration and substitution (A1)
532 A1
3
Question 6Structured · 7 marks

A continuous 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]
(c)

Find the median of X.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)38
∫ f(x) dx = 1 (M1)
38 A1
2
(b)32
∫ x f(x) dx (M1)
32 A1
2
(c)1.59
∫ from 0 to m = 0.5 (M1)
m3 = 82 (A1)
1.59 A1
3

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Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in