Continuous random variables — AS & A Level Further Mathematics (9231) questions by topic
Probability density and cumulative distribution work fills this topic: finding the cumulative distribution function for a two-piece exponential-form density, sketching a piecewise cubic-and-linear density function, and showing that a normalising constant k equals 6/17 for a two-piece density.
Find, show and sketch are the top command words across 16 questions from 2023 to 2025 papers, worth from 8 to 12 marks, where handling piecewise definitions correctly at their boundary is a recurring hurdle. Practising each integration and sketch with instant marking catches an omitted constant of integration or an incorrectly combined tail condition before the same gap resurfaces in a later density-function question.
What examiners look for
- A recurring criticism is omitting the constant of integration when finding F(x).
- Examiners note incorrectly combining the tails as 'F(x) = 0 otherwise' instead of stating F(x) = 0 below the range and F(x) = 1 above it.
- A common mistake offered the second, out-of-range root of a quadratic alongside the valid answer.
- Reports flag deriving only one equation in a and b, usually from integrating, without the second equation from equating the pieces at the boundary.
- Strong scripts worked with the probability density function throughout, derived the constant via a definite integral or area argument, and obtained two independent equations in a and b.
May–June 2025, Paper 41
The diagram shows the graph of a probability density function f(x). The curve is constant (a horizontal line at height f = a) for 0 <= x <= 5, then decreases linearly from x = 5…
Answer this question and get it marked →May–June 2025, Paper 42
As shown in the diagram, the continuous random variable X has probability density function f given by f(x) = a for 0 <= x <= 5, f(x) = b - cx for 5 <= x <= 8, f(x) = 0 otherwise,…
Answer this question and get it marked →May–June 2025, Paper 43
A continuous random variable X has probability density function f given by f(x) = kx for 0 <= x < 1, f(x) = k(8 - x) for 1 <= x <= 8, f(x) = 0 otherwise, where k is a constant.…
Answer this question and get it marked →May–June 2025, Paper 44
The continuous random variable X has probability density function f given by f(x) = kx for 0 <= x < 1, f(x) = kx^2 for 1 <= x <= 2, f(x) = 0 otherwise. Show that k = 6/17. [2]
Answer this question and get it marked →May–June 2024, Paper 41
The continuous random variable X has probability density function f given by f(x) = (x/4)(4 - x^2) for 0 <= x <= 2, f(x) = 0 otherwise. (a) Find Var(sqrt(X)). [4]
Answer this question and get it marked →May–June 2024, Paper 42
The continuous random variable X has probability density function f given by f(x) = (x/4)(4 - x^2) for 0 <= x <= 2, f(x) = 0 otherwise. Find Var(sqrt(X)). [4]
Answer this question and get it marked →May–June 2024, Paper 43
The continuous random variable X has cumulative distribution function F given by F(x) = 0 for x < 2, F(x) = (x - 2)^2 / 12 for 2 <= x < 4, F(x) = 1 - (8 - x)^2 / 24 for 4 <= x <=…
Answer this question and get it marked →October–November 2024, Paper 41
The continuous random variable X has probability density function f given by f(x) = kx^3 for 0 <= x < 1, f(x) = k(5 - x) for 1 <= x <= 5, f(x) = 0 otherwise, where k is a…
Answer this question and get it marked →October–November 2024, Paper 42
The random variable X has probability density function f given by f(x) = (1/21)(x - 1)^2 for 2 <= x <= 5, f(x) = 0 otherwise. Find the cumulative distribution function of X. [3]
Answer this question and get it marked →October–November 2024, Paper 43
The continuous random variable X has probability density function f given by f(x) = k x^3 for 0 <= x < 1, f(x) = k(5 - x) for 1 <= x <= 5, f(x) = 0 otherwise, where k is a…
Answer this question and get it marked →May–June 2023, Paper 41
The continuous random variable X has probability density function f given by f(x) = (3/28)(e^(x/2) + 4e^(-x/2)) for 0 <= x <= 2 ln 3, f(x) = 0 otherwise. (a) Find the cumulative…
Answer this question and get it marked →May–June 2023, Paper 42
The continuous random variable X has probability density function f given by f(x) = (3/28)(e^(x/2) + 4e^(-x/2)) for 0 <= x <= 2 ln 3, f(x) = 0 otherwise. Find the cumulative…
Answer this question and get it marked →May–June 2023, Paper 43
The continuous random variable X has probability density function f given by f(x) = (1/6)(x^(-1/3) - x^(-2/3)) for 1 <= x <= 27, f(x) = 0 otherwise. Find the cumulative…
Answer this question and get it marked →October–November 2023, Paper 41
As shown in the diagram, the continuous random variable X has probability density function f given by f(x) = m x for 0 <= x <= 2, f(x) = k/x^2 + c for 2 <= x <= 6, f(x) = 0…
Answer this question and get it marked →October–November 2023, Paper 42
The diagram shows the continuous random variable X with probability density function f given by f(x) = (1/128)(4ax - bx^3) for 0 <= x <= 4, f(x) = c for 4 <= x <= 6, f(x) = 0…
Answer this question and get it marked →October–November 2023, Paper 43
As shown in the diagram, the continuous random variable X has probability density function f given by f(x) = mx for 0 <= x <= 2, f(x) = k/x^2 + c for 2 <= x <= 6, f(x) = 0…
Answer this question and get it marked →Other AS & A Level Further Mathematics topics
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