Probability generating functions: past paper questions

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• Question 6May/June 2026, paper 41Structured10 marks

  The discrete random variable $X$ has probability generating function ${\mathrm{G}}_X(t)$ defined by $${\mathrm{G}}_X(t)=\frac{k}{t}(at+1{)}^2\mathrm{.}$$ It is given that $\mathrm{E}(X)=\frac{1}{3}$. Show that $a=2$ and find the value of $k$.

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• Question 7Oct/Nov 2025, paper 41Structured10 marks

  A discrete random variable $X$ takes values $r=0,1,2$ with probabilities $\mathrm{P}(X=r)$ as given in the following table.

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• Question 5Oct/Nov 2024, paper 41Structured9 marks

  Nikita has three coins. One coin is fair, one coin is biased so that the probability of obtaining a head is $\frac{1}{3}$ and the third coin is biased so that the probability of obtaining a head is $\frac{1}{5}$.

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• Question 5Oct/Nov 2023, paper 41Structured10 marks

  The random variable $X$ has the geometric distribution $\mathrm{G}\mathrm{e}\mathrm{o}(p)$. Show that the probability generating function of $X$ is $\frac{pt}{1-qt}$, where $q=1-p$.

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• Question 6May/June 2026, paper 42Structured10 marks

  The discrete random variable $X$ has probability generating function ${\mathrm{G}}_X(t)$ defined by $${\mathrm{G}}_X(t)=\frac{k}{t}(at+1{)}^2\mathrm{.}$$ It is given that $\mathrm{E}(X)=\frac{1}{3}$. Show that $a=2$ and find the value of $k$.

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• Question 6Oct/Nov 2025, paper 42Structured10 marks

  The discrete random variable $X$ has probability generating function ${\mathrm{G}}_X(t)$ given by $${\mathrm{G}}_X(t)=\frac{t}{(3-2t{)}^2}\mathrm{.}$$ Find $\mathrm{E}(X)$ and $\mathrm{V}\mathrm{a}\mathrm{r}(X)$.

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• Question 2Oct/Nov 2024, paper 42Structured8 marks

  The random variable $X$ has probability generating function ${\mathrm{G}}_X(t)$ given by $${\mathrm{G}}_X(t)=\frac{1}{5}+pt+q{t}^2,$$ where $p$ and $q$ are constants. Given that $\mathrm{E}(X)=1.1$, find the numerical value of $\mathrm{Var}(X)$.

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• Question 3Oct/Nov 2023, paper 42Structured10 marks

  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 marbles that Toby obtains.

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• Question 3May/June 2026, paper 43Structured7 marks

  The discrete random variable $X$ has probability generating function ${\mathrm{G}}_X(t)$ given by $${\mathrm{G}}_X(t)=0.1+0.3t+0.4t^2+0.2t^3\mathrm{.}$$ Use ${\mathrm{G}}_X(t)$ to find $\mathrm{E}(X)$ and $\mathrm{Var}(X)$.

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• Question 5Oct/Nov 2023, paper 43Structured10 marks

  The random variable $X$ has the geometric distribution $\mathrm{G}\mathrm{e}\mathrm{o}(p)$. Show that the probability generating function of $X$ is $\frac{pt}{1-qt}$, where $q=1-p$.

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• Question 6May/June 2025, paper 41Structured6 marks

  $Y$ is a discrete random variable which takes the values $0,2,4,\dots$.

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• Question 4May/June 2024, paper 41Structured7 marks

  The random variable $Y$ is the sum of two independent observations of the random variable $X$. The probability generating function $G_Y(t)$ of $Y$ is given by $$G_Y(t)=\frac{t^2}{(4-3t{)}^4}\mathrm{.}$$ Find $\mathrm{E}(Y)$.

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• Question 5May/June 2023, paper 41Structured9 marks

  Harry has three coins.

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• Question 6May/June 2025, paper 42Structured6 marks

  Y is a discrete random variable which takes the values 0, 2, 4, … . The probability generating function of Y is given by G_Y(t) = k/(1 - at²). Find k in terms of a.

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• Question 5May/June 2023, paper 42Structured9 marks

  Harry has three coins. • One coin is biased so that, when it is thrown, the probability of obtaining a head is $\frac{1}{3}$. • The second coin is biased so that, when it is thrown, the probability of obtaining a head is $\frac{1}{4}$.

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• Question 6May/June 2025, paper 43Structured11 marks

  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 different colours present in Kieran’s selection is denoted by $Y$.

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• Question 4May/June 2024, paper 43Structured9 marks

  The random variable $X$ has probability generating function $G_X(t)$ given by $$G_X(t)=ct(1+t{)}^5,$$ where $c$ is a constant. Find the value of $c$.

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• Question 5May/June 2023, paper 43Structured9 marks

  The random variable $X$ has probability generating function ${\mathrm{G}}_X(t)$ given by $${\mathrm{G}}_X(t)=k(1+3t+4t^2),$$ where $k$ is a constant. Show that $\mathrm{E}(X)=\frac{11}{8}$.

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