Linear combinations of random variables: past paper questions

Where the questions come from

• Question 1May/June 2026, paper 61Structured5 marks

  The masses, in grams, of apples of a certain type have the distribution $\mathrm{N}(58.4,{6.3}^2)$. The apples are packed in bags, with each bag containing 10 randomly chosen apples.

  Try this question
• Question 1Oct/Nov 2025, paper 61Structured9 marks

  The random variables $X$ and $Y$ have independent distributions $X\sim \mathrm{P}\mathrm{o}(3)$ and $Y\sim \mathrm{P}\mathrm{o}(2)$ respectively. Find $\mathrm{P}(2<X<5)$.

  Try this question
• Question 2Oct/Nov 2024, paper 61Structured5 marks

  The masses, in kilograms, of small and large bags of wheat have the independent distributions $\mathrm{N}(16.0,0.4)$ and $\mathrm{N}(51.0,0.9)$ respectively.

  Try this question
• Question 4Oct/Nov 2023, paper 61Structured8 marks

  The masses, in kilograms, of chemicals $A$ and $B$ produced per day by a factory are modelled by the independent random variables $X$ and $Y$ respectively, where $X\sim \mathrm{N}(10.3,5.76)$ and $Y\sim \mathrm{N}(11.4,9.61)$.

  Try this question
• Question 5May/June 2026, paper 61Structured10 marks

  $X$ and $Y$ are independent random variables such that $X\sim \mathrm{P}\mathrm{o}(2.5)$ and $Y\sim \mathrm{P}\mathrm{o}(\lambda )$. Find $\mathrm{P}(1⩽X⩽3)$.

  Try this question
• Question 6Oct/Nov 2025, paper 61Structured10 marks

  The masses, in kilograms, of large and small bags of potatoes have the independent distributions $\mathrm{N}(2.5,0.05)$ and $\mathrm{N}(0.8,0.02)$ respectively.

  Try this question
• Question 5Oct/Nov 2024, paper 62Structured11 marks

  A machine puts sweets into bags at random. The numbers of lemon and orange sweets in a bag have the independent distributions $\mathrm{P}\mathrm{o}(3.7)$ and $\mathrm{P}\mathrm{o}(2.6)$ respectively. A bag of sweets is chosen at random.

  Try this question
• Question 6Oct/Nov 2023, paper 62Structured7 marks

  A factory makes loaves of bread in batches. One batch of loaves contains $X$ kilograms of dried yeast and $Y$ kilograms of flour, where $X$ and $Y$ have the independent distributions $\mathrm{N}(0.7,{0.02}^2)$ and $\mathrm{N}(100.0,{3.0}^2)$ respectively.

  Try this question
• Question 2May/June 2026, paper 62Structured5 marks

  Each week, Birgitte takes three train journeys. The times, in minutes, for these journeys have the independent distributions $A\sim \mathrm{N}(28,15)$, $B\sim \mathrm{N}(23,8)$ and $C\sim \mathrm{N}(99,24)$. For a randomly chosen week, find $\mathrm{P}(C>2(A+B))$.

  Try this question
• Question 2Oct/Nov 2025, paper 62Structured6 marks

  The random variable $X$ has a normal distribution with mean 10 and standard deviation 3. The independent random variable $Y$ has a Poisson distribution with mean 4. Find the standard deviation of $X+Y$.

  Try this question
• Question 4May/June 2026, paper 62Structured10 marks

  The random variables $D$ and $E$ have the independent distributions $D\sim \mathrm{P}\mathrm{o}(1.7)$ and $E\sim \mathrm{P}\mathrm{o}(2.1)$. Find $\mathrm{P}(D+E<4)$.

  Try this question
• Question 2May/June 2024, paper 61Structured5 marks

  The random variable $X$ has the distribution $\mathrm{N}(31.2,{10.4}^2)$. Two independent random values of $X$, denoted by $X_1$ and $X_2$, are chosen. Find $\mathrm{P}(X_1>3X_2)$.

  Try this question
• Question 5May/June 2023, paper 61Structured7 marks

  Large packets of rice are packed in cartons, each containing 20 randomly chosen packets.

  Try this question
• Question 6May/June 2026, paper 63Structured10 marks

  One batch of a cake mixture contains $X\text{ litres}$ of ingredient $A$ and $Y\text{ litres}$ of ingredient $B$, where $X$ and $Y$ have the independent distributions $X\sim \mathrm{N}(25.5,0.07)$ and $Y\sim \mathrm{N}(26.0,0.08)$.

  Try this question
• Question 4May/June 2024, paper 62Structured6 marks

  A random variable $X$ has the distribution $\mathrm{N}(10,12)$. Two independent values of $X$, denoted by $X_1$ and $X_2$, are chosen at random. Write down the value of $\mathrm{P}(X_1>X_2)$.

  Try this question
• Question 5May/June 2023, paper 62Structured11 marks

  Two random variables $X$ and $Y$ have the independent distributions $\mathrm{N}(7,3)$ and $\mathrm{N}(6,2)$ respectively. A random value of each variable is taken. Find the probability that the two values differ by more than 2.

  Try this question
• Question 4May/June 2025, paper 61Structured6 marks

  At an entertainment centre, the cost for using a particular video game is $0.40 per minute. The number of minutes for which people use the video game has mean 15 and variance 9. Find the mean and variance of the amount people pay for using the video game.

  Try this question
• Question 3May/June 2024, paper 63Structured5 marks

  The masses in kilograms of large and small bags of cement have the independent distributions $\mathrm{N}(50,2.4)$ and $\mathrm{N}(26,1.8)$ respectively.

  Try this question
• Question 4May/June 2023, paper 63Structured6 marks

  The mass, in tonnes, of steel produced per day at a factory is normally distributed with mean 65.2 and standard deviation 3.6. It can be assumed that the mass of steel produced each day is independent of other days.

  Try this question
• Question 5May/June 2025, paper 61Structured11 marks

  The random variables $W$ and $X$ have the independent distributions $\mathrm{P}\mathrm{o}(1.2)$ and $\mathrm{P}\mathrm{o}(2.3)$ respectively. Find $\mathrm{P}(3⩽W+X⩽5)$.

  Try this question
• Question 7May/June 2024, paper 63Structured13 marks

  The independent random variables $X$ and $Y$ have the distributions $\mathrm{P}\mathrm{o}(1.9)$ and $\mathrm{P}\mathrm{o}(2.2)$ respectively. Find $\mathrm{P}(X+Y<4)$.

  Try this question
• Question 1May/June 2025, paper 62Structured6 marks

  One of a group of three students is to be chosen at random. Explain how a single throw of a fair six-sided dice could be used to make the choice.

  Try this question
• Question 6May/June 2025, paper 62Structured10 marks

  Use suitable approximating distributions to answer the following. The random variable $W$ has the distribution $\text{B}(700,0.005)$. Find $\text{P}(W⩾4)$.

  Try this question
• Question 5May/June 2025, paper 63Structured4 marks

  In Urberia, the masses, in kilograms, of men have the distribution $\mathrm{N}(70.3,{5.9}^2)$. A certain footbridge in Urberia can take a maximum safe load of $1500\text{ kg}$. When $n$ men stand on the bridge, the probability that the bridge is unsafe is less than 0.01.

  Try this question