Probability density functions: AQA A‑Level Further Maths questions

AQA A-Level Further Maths past-paper questions on SD.1 Probability density functions, 2019–2025, with mark schemes and examiner’s reports where available. Add MadAsMaths questions to your practice, or browse them separately by topic and difficulty.

All 19 questions on Probability density functions, as a list

June 2025, Paper 3 Statistics, Q1 (1 mark): The continuous random variable X has standard deviation 16 Find Var(5X – 12) Circle your answer. [1 mark] 80 400 6388 6400
June 2025, AS Paper 2 Statistics, Q1 (1 mark): The continuous random variable A has probability density function f (a) and variance 16 It is given that f (a) = 0 for a < 0 and ∫ ∞ 0 a2f (a)da = 52 Find the m
June 2025, AS Paper 2 Statistics, Q4 (5 marks): The discrete random variable X has probability distribution x 1 2 a P(X = x) 0.2 0.7 0.1 where a is a constant greater than 2 4 (a) Find P(X ≥ 2) [1 mark] 4 (b)
June 2025, AS Paper 2 Statistics, Q6 (6 marks): The continuous random variable A represents the score achieved by a player in a particular game. The probability density function of A is f (a) The probability
June 2024, Paper 3 Statistics, Q5 (5 marks): The continuous random variable X has probability density function f(x) = 1 6 e x 3 0 ≤ x ≤ In 27 0 otherwise { Show that the mean of X is 3 2 (In 27 – 2) [5 mar
June 2024, AS Paper 2 Statistics, Q6 (11 marks): The continuous random variable X has probability density function f(x) = 3x 44 + 1 22 1 ≤ x ≤ 5 0 otherwise { 6 (a) Find P(X > 2) [2 marks] 6 (b) Find the upper
June 2023, Paper 3 Statistics, Q8 (14 marks): The continuous random variable X has probability density function f (x) = k sin 2x 0 − x − p 6 0 otherwise 8 < : where k is a constant. 8 (a) Show that k = 4 [4
June 2023, AS Paper 2 Statistics, Q1 (1 mark): The continuous random variable X has variance 9 The discrete random variable Y has standard deviation 2 and is independent of X Find Var(X + Y ) Circle your ans
June 2023, AS Paper 2 Statistics, Q8 (8 marks): The continuous random variable X has probability density function f (x) It is given that f (x) = x2 for 0 ≤ x ≤ 1 It is also given that f (x) is a linear functi
June 2022, Paper 3 Statistics, Q2 (1 mark): The random variable X has probability density function f (x) = 1 0 < x − 1 2 3 8 x×2 1 2 < x − 3 2 0 otherwise 8 > > > > < > > > > : Find P(X < 1) Circle your a
June 2022, Paper 3 Statistics, Q8 (11 marks): The continuous random variable X has cumulative distribution function F(x) where F(x) = 0 x = 0 ekx × 1 0 − x − 5 1 x > 5 8 > > < > > : 8 (a) Show that k = 1 5
June 2022, AS Paper 2 Statistics, Q2 (1 mark): The continuous random variable Y has probability density function f (y) where (1 −1 yf (y) dy = 16 and (1 −1 y2f (y) dy = 1040 Find the standard deviation of Y
June 2022, AS Paper 2 Statistics, Q5 (11 marks): The continuous random variable X has probability density function f (x) = x3 0 < x × 1 9 1696x3(x2 + 1) 1 < x × 3 0 otherwise 8 > > > < > > > : 5 (a) Find P(X <
November 2021, Paper 3 Statistics, Q5 (6 marks): The continuous random variable X has cumulative distribution function F(x) = 0 x × 1 1 10x × 1 10 1 < x × 6 1 90x2 + 1 10 6 < x × 9 1 x > 9 8 > > > > > > > < >
November 2021, AS Paper 2 Statistics, Q6 (11 marks): The continuous random variable X has probability density function f (x) = 1 114 (4x + 7) 0 × x × 6 0 otherwise 8 < : 6 (a) Show that the median of X is 3.87, co
November 2020, Paper 3 Statistics, Q1 (1 mark): The continuous random variable X has probability density function f (x) = 1 5 1 − x − 6 0 otherwise 8 > < > : Find P(X × 3) Circle your answer. [1 mark] 1 5 2 5
November 2020, AS Paper 2 Statistics, Q6 (8 marks): The continuous random variable X has probability density function f (x) = 4 45 (x3 × 10x2 + 29x × 20) 1 × x × 4 0 otherwise 8 < : 6 (a) Find P(X < 2) [2 marks]
June 2019, Paper 3 Statistics, Q5 (7 marks): An insurance company models the claims it pays out in pounds (£) with a random variable X which has probability density function f (x) = k x 1 < x < a 0 otherwi
June 2019, AS Paper 2 Statistics, Q4 (8 marks): The continuous random variable X has probability density function f (x) = 4 99(12x − x2 − x3) 0 × x × 3 0 otherwise 8 < : 4 (a) Find P(X > 1) [3 marks] 4 (b) Sh

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