Continuous random variables — AS & A Level Mathematics (9709) questions by topic
Proving a probability density function's constant from the requirement that total probability equals one, using symmetry in a distribution to compare two probability regions, and finding an expectation from a given density function are typical of the 17 continuous random variables questions from 2023 to 2025, worth 3 to 11 marks.
Show, find, explain and state are the command words tracked here, with show demanding a fully justified integration rather than a stated constant. Because a probability density function must integrate to exactly one, submit your integration working for instant marking, so a wrong limit or a missed factor in the constant is caught immediately rather than undermining every later part of the question.
May–June 2025, Paper 61
X is a random variable with probability density function given by f(x) = 1 + cos(pi x) for 0 <= x <= 1, f(x) = 0 otherwise. Show that P(X < 1/2) = 1/2 + 1/pi.
Answer this question and get it marked →May–June 2025, Paper 62
The random variable X has probability density function given by f(x) = kx^2 / a^2 for 0 <= x <= a, f(x) = 0 otherwise, where k and a are positive constants. Show that k = 3/a.
Answer this question and get it marked →May–June 2025, Paper 63
A random variable X has probability density function given by f(x) = ax for 0 <= x <= b, f(x) = 0 otherwise, where a and b are constants. Show that a = 2/b^2.
Answer this question and get it marked →May–June 2025, Paper 65
The diagram shows the graph of the probability density function f of a random variable X. Between x = 0 and x = a the graph consists of a straight line through O with gradient k,…
Answer this question and get it marked →May–June 2024, Paper 61
The diagram shows the graph of the probability density function, f, of a random variable X. The graph is a quarter circle entirely in the first quadrant with centre (0, 0) and…
Answer this question and get it marked →May–June 2024, Paper 62
The probability density function, f, of a random variable X is given by f(x) = k(1 + cos x) for 0 <= x <= pi, f(x) = 0 otherwise, where k is a constant. Show that k = 1/pi.
Answer this question and get it marked →May–June 2024, Paper 63
A random variable X has probability density function f given by f(x) = ax - x^3 for 0 <= x <= sqrt(2), and f(x) = 0 otherwise, where a is a constant. Show that a = 2.
Answer this question and get it marked →October–November 2024, Paper 61
A random variable X has probability density function f defined by f(x) = a/x^2 - 18/x^3 for 2 <= x <= 3, and f(x) = 0 otherwise, where a is a constant. (a) Show that a = 27/2.
Answer this question and get it marked →October–November 2024, Paper 62
The time, X hours, taken by a large number of people to complete a challenge is modelled by the probability density function given by f(x) = 1/x² for a ≤ x ≤ b, and f(x) = 0…
Answer this question and get it marked →October–November 2024, Paper 63
A random variable X has probability density function f defined by f(x) = a/x^2 - 18/x^3 for 2 <= x <= 3, and f(x) = 0 otherwise, where a is a constant. Show that a = 27/2.
Answer this question and get it marked →May–June 2023, Paper 61
The graph of the function f is a straight line segment from (0, 0) to (2, 1). (On the diagram, f(x) is plotted against x, with the value 1 marked on the f(x) axis and the values…
Answer this question and get it marked →May–June 2023, Paper 62
The diagram shows the graph of the probability density function, f, of a random variable X which takes values between 0 and 4 only. Between these two values the graph is a…
Answer this question and get it marked →May–June 2023, Paper 63
A random variable X has probability density function f, where f(x) = (3/2)(1 - x^2) for 0 <= x <= 1, and f(x) = 0 otherwise. Find E(X).
Answer this question and get it marked →A random variable X has probability density function f, where the graph of y = f(x) is a semicircle with centre (0, 0) and radius sqrt(2/pi), entirely above the x-axis. Elsewhere…
Answer this question and get it marked →October–November 2023, Paper 61
A continuous random variable X takes values from 0 to 6 only and has a probability distribution that is symmetrical. Two values, a and b, of X are such that P(a < X < b) = p and…
Answer this question and get it marked →October–November 2023, Paper 62
The random variable X has probability density function f given by f(x) = 1/x² for a < x < b, and f(x) = 0 otherwise, where a and b are positive constants. It is given that E(X) =…
Answer this question and get it marked →October–November 2023, Paper 63
A continuous random variable X takes values from 0 to 6 only and has a probability distribution that is symmetrical. Two values, a and b, of X are such that P(a < X < b) = p and…
Answer this question and get it marked →Other AS & A Level Mathematics topics
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