Probability generating functions — AS & A Level Further Mathematics (9231) questions by topic
Manipulating probability generating functions defines this topic: finding a constant k for a discrete variable taking even values, combining the pgfs of three biased coins, and differentiating a given pgf to find E(Y) for a sum of two independent observations.
Find, show, write, state and explain are the top command words across 16 questions from 2023 to 2025 papers, worth 6 to 11 marks, where multiplying rather than adding two pgfs is the pivotal step. Checking each pgf derivation and differentiation with instant marking catches a numerical slip in multiplying generating functions, or an incorrect power of t, before the same error carries into a variance calculation.
What examiners look for
- A recurring criticism is a few candidates using sampling with replacement instead of choosing without replacement.
- Examiners note very occasionally the two probability generating functions were added instead of multiplied.
- A common mistake paired correct probabilities with incorrect powers of t in the generating function.
- Strong scripts used the correct method to obtain the correct probability generating function, and understood that the pgf from an earlier part had to be multiplied by the given pgf for Y.
- Candidates who differentiated the probability generating function and substituted t = 1 correctly obtained E(Z).
May–June 2025, Paper 41
Y is a discrete random variable which takes the values 0, 2, 4, ... . The probability generating function of Y is given by GY(t) = k / (1 - at^2). Find k in terms of a.
Answer this question and get it marked →May–June 2025, Paper 42
Y is a discrete random variable which takes the values 0, 2, 4, ... . The probability generating function of Y is given by GY(t) = k / (1 - a t^2). Find k in terms of a. [1]
Answer this question and get it marked →May–June 2025, Paper 43
A bag contains 7 red balls and 3 blue balls. Kieran selects 2 balls at random, without replacement. The number of red balls selected by Kieran is denoted by X, and the number of…
Answer this question and get it marked →May–June 2025, Paper 44
Eric has three identical coins, each of which is biased so that the probability of obtaining a head when it is thrown is 1/3. The random variable X is the number of heads obtained…
Answer this question and get it marked →May–June 2024, Paper 41
The random variable Y is the sum of two independent observations of the random variable X. The probability generating function GY(t) of Y is given by GY(t) = t^2 / (4 - 3t)^4. (a)…
Answer this question and get it marked →May–June 2024, Paper 42
The random variable Y is the sum of two independent observations of the random variable X. The probability generating function GY(t) of Y is given by GY(t) = t^2 / (4 - 3t)^4.…
Answer this question and get it marked →May–June 2024, Paper 43
The random variable X has probability generating function GX(t) given by GX(t) = ct(1 + t)^5, where c is a constant. Find the value of c.
Answer this question and get it marked →October–November 2024, Paper 41
Nikita has three coins. One coin is fair, one coin is biased so that the probability of obtaining a head is 1/3 and the third coin is biased so that the probability of obtaining a…
Answer this question and get it marked →October–November 2024, Paper 42
The random variable X has probability generating function GX(t) given by GX(t) = 1/5 + p t + q t^2, where p and q are constants. Given that E(X) = 1.1, find the numerical value of…
Answer this question and get it marked →October–November 2024, Paper 43
Nikita has three coins. One coin is fair, one coin is biased so that the probability of obtaining a head is 1/3 and the third coin is biased so that the probability of obtaining a…
Answer this question and get it marked →May–June 2023, Paper 41
Harry has three coins. - One coin is biased so that, when it is thrown, the probability of obtaining a head is 1/3. - The second coin is biased so that, when it is thrown, the…
Answer this question and get it marked →May–June 2023, Paper 42
Harry has three coins. - One coin is biased so that, when it is thrown, the probability of obtaining a head is 1/3. - The second coin is biased so that, when it is thrown, the…
Answer this question and get it marked →May–June 2023, Paper 43
The random variable X has probability generating function GX(t) given by GX(t) = k(1 + 3t + 4t^2), where k is a constant. Show that E(X) = 11/8.
Answer this question and get it marked →October–November 2023, Paper 41
The random variable X has the geometric distribution Geo(p). (a) Show that the probability generating function of X is (p t)/(1 - q t), where q = 1 - p.
Answer this question and get it marked →October–November 2023, Paper 42
Toby has a bag which contains 6 red marbles and 3 green marbles. He randomly chooses 3 marbles from the bag, without replacement. The random variable X is the number of red…
Answer this question and get it marked →October–November 2023, Paper 43
The random variable X has the geometric distribution Geo(p). Show that the probability generating function of X is pt/(1 - qt), where q = 1 - p.
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