Linear combinations of random variables

The twelve values of $x$ are listed here: $1761.6,\;1758.5,\;1762.3,\;1761.4,\;1759.4,\;1759.1,\;1762.5,\;1761.9,\;1762.4,\;1761.9,\;1762.8,\;1761.0$.

Feb/March 2017

Pack A contains ten cards labelled $0, 0, 1, 1, 1, 1, 1, 3, 3, 3$. Pack B contains six cards labelled $0, 0, 2, 2, 2, 2$. One card is drawn at random from each pack. The random variable $X$ is the total of the two card numbers.

Feb/March 2017

The discrete random variable $X$ is described by the probability distribution shown below.

Feb/March 2018

For large and small cups of tea, the volumes in millilitres are represented by the distributions $N(200, 30)$ and $N(110, 20)$, respectively.

Feb/March 2020

The juice volumes, measured in litres, for large bottles and small bottles have distributions $N(5.10, 0.0102)$ and $N(2.51, 0.0036)$ respectively.

Feb/March 2021

The building heights in a large city are modeled by a normal distribution with mean $18.3\,\text{m}$ and standard deviation $2.5\,\text{m}$.

Feb/March 2022

For large and small packets of Maxwheat cereal, the masses in grams are independently distributed as $N(410.0,\,3.6^2)$ and $N(206.0,\,3.7^2)$, respectively.

Feb/March 2023

Each year, a transport company consumes $X$ litres of gasoline and $Y$ litres of diesel fuel, with $X$ and $Y$ independent and distributed as $X \sim N(10\,700,\,950^2)$ and $Y \sim N(13\,400,\,1210^2)$.

Feb/March 2024

The random variables $X$ and $Y$ are independent, with distributions $N(44, 16)$ and $N(30, 9)$ respectively.

Feb/March 2025

The distribution of the random variable $X$ is $\operatorname{Po}(1.5)$.

Feb/March 2025

The table below displays how many rides two students, Fei and Graeme, went on at a fairground.

May/June 2010

Two fair twelve-sided dice numbered $1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12$ are rolled, and the numbers on the faces that land face down are recorded. Events $Q$ and $R$ are defined below. $Q$: the product of the two numbers is $24$. $R$: both of the numbers are greater than $8$.

May/June 2010

A farm keeps $5$ ducks and $2$ geese. Four birds are selected at random. The random variable $X$ gives the number of geese selected.

May/June 2010

Set $A$ contains ten digits: $0, 0, 0, 0, 0, 0, 2, 2, 2, 4$. Set $B$ contains seven digits: $0, 0, 0, 0, 2, 2, 2$. A digit is selected at random from each set. The random variable $X$ is the total of the two selected digits.

May/June 2010

Tim rolls a fair die two times and records the number on each roll.

May/June 2011

A spinner has $5$ equal sectors, labelled $1$, $2$, $3$, $4$ and $5$. The number on the sector where it stops is taken as the score. This score is represented by the random variable $X$, and its probability distribution is listed in the table: $P(X=1)=0.3$, $P(X=2)=0.15$, $P(X=3)=3p$, $P(X=4)=2p$, $P(X=5)=0.05$. There is also a second spinner with $3$ sectors, labelled $1$, $2$ and $3$. The score from this spinner is represented by the random variable $Y$. It is known that $P(Y=1)=0.3$, $P(Y=2)=0.5$ and $P(Y=3)=0.2$.

May/June 2012

The random variable $X$ follows the probability distribution given in the table: $x = 2, 4, 6$ with $P(X = x) = 0.5, 0.4, 0.1$. Two independent values of $X$ are selected at random. The random variable $Y$ is $0$ when the two values of $X$ match. Otherwise, $Y$ equals the larger value of $X$ minus the smaller value of $X$.

May/June 2012

A fair die has faces marked $1, 1, 2, 2, 3, 3$. Let the random variable $W$ denote the score on one throw, namely the number showing on the top face after the die comes to rest.

May/June 2012

Susan has a bag of sweets with 7 chocolates and 5 toffees inside it. Ahmad has a bag of sweets with 3 chocolates, 4 toffees and 2 boiled sweets inside it. One sweet is selected at random from Susan’s bag and added to Ahmad’s bag. Then one sweet is chosen at random from Ahmad’s bag.

May/June 2013

Roger and Andy are playing a tennis match; the match is won by the first player to take two sets. The chance that Roger wins the opening set is $0.6$. After the first set, Roger's probability of winning a set is $0.7$ if he won the previous set, and is $0.25$ if he lost the previous set. No set is drawn.

May/June 2014

Coin $A$ has been weighted so that the probability of a head is $\frac{2}{3}$. Coin $B$ has been weighted so that the probability of a head is $\frac{1}{4}$. Coin $A$ is tossed twice and coin $B$ is tossed once.

May/June 2014

The table presents the mean and standard deviation of the weights of some turkeys and geese.

May/June 2015

A certain species of bird produces 1, 2, 3 or 4 eggs in a nest each year. The probability of $x$ eggs is $kx$, where $k$ is a constant.

May/June 2016

Two fair ordinary dice are rolled. The score is determined in this way: if the numbers shown are different, the score is the smaller number; if both dice show the same number, the score is $0$.

May/June 2016

In this probability distribution, the random variable $X$ takes the value $x$ with probability $kx^2$, where $k$ is a constant, and the only values that $x$ can have are $-2, -1, 2, 4$.

May/June 2017

Two matching biased triangular spinners, each with sides labelled $1$, $2$ and $3$, are spun. For each spinner, the chances of landing on the sides marked $1$, $2$ and $3$ are $p$, $q$ and $r$ respectively. The score is the total of the numbers on the sides where the spinners land. You are given that $P(\text{score is } 6) = \frac{1}{36}$ and $P(\text{score is } 5) = \frac{1}{9}$.

May/June 2017

During the school holidays, on any day Khalid either cycles with probability $0.6$, or rides his skateboard with probability $0.4$. He never does both on the same day. If he cycles, the probability that he hurts himself is $0.05$. If he uses his skateboard, the probability that he hurts himself is $0.75$.

May/June 2017

A loaded die has six faces, labelled $1$ to $6$. For this die, the chance of getting $1$, $3$ or $5$ is $0.1$ each. The chance of getting $2$ or $4$ is $0.2$ each. The die is rolled two times.

May/June 2017

Mrs Rupal selects $3$ animals at random from $5$ dogs and $2$ cats. Let the random variable $X$ represent the number of cats selected.

May/June 2018

A game uses $3$ coins, $A$, $B$ and $C$. Coins $A$ and $B$ are biased so that the chance of getting a head is $0.4$ for coin $A$ and $0.75$ for coin $B$. Coin $C$ is unbiased. The $3$ coins are tossed once.

May/June 2018

A fair six-sided die is rolled two times, and the outcomes are recorded. Event $X$ is defined as ‘The sum of the two scores is 4’. Event $Y$ is defined as ‘The first score is 2 or 5’.

May/June 2019

At a funfair, Amy pays $1$ for two tries at making a bell ring by firing a water pistol at it. • If the bell rings on her first try, she is paid $3$ and stops playing. So, in total, her gain is $2$. • If the bell rings on her second try, she is paid $1.50$ and stops playing. So, in total, her gain is $0.50$. • If the bell does not ring in either of the two tries, she loses her original $1$. The chance that Amy makes the bell ring on any try is $0.2$, independently of the other tries.

May/June 2019

Maryam has $7$ sweets in a tin, consisting of $6$ toffees and $1$ chocolate. She picks one sweet at random and removes it. Her friend then puts $3$ chocolates into the tin. After that, Maryam selects another sweet at random from the tin.

May/June 2019

A fair spinner with five sides has faces numbered $1, 1, 1, 2, 3$. A fair spinner with three sides has faces numbered $1, 2, 3$. The two spinners are each spun once, and the score is found by multiplying the numbers shown.

May/June 2019

The independent distributions of the masses, measured in kilograms, of large sacks of flour and small sacks of flour are $N(40, 1.5^2)$ and $N(12, 0.7^2)$ respectively.

May/June 2020

At the gym each day, Sarah runs three times. The three run distances, measured in metres, are independently distributed as $W \sim \mathcal{N}(1520, 450)$, $X \sim \mathcal{N}(2250, 720)$ and $Y \sim \mathcal{N}(3860, 1050)$.

May/June 2020

The weights, in kilograms, of large sacks and small sacks of flour are distributed as $N(55, 3^2)$ and $N(27, 2.5^2)$ respectively.

May/June 2021

The number of goals a team scores in a match is independent of any other matches, and the random variable $X$ represents this. $X$ has a Poisson distribution with mean $1.36$. A supporter says they will donate $5 to the team for every goal the team scores in the next $10$ matches.

May/June 2021

The lengths, measured in centimetres, of two insect types, $A$ and $B$, are represented by the random variables $X \sim N(6.2, 0.36)$ and $Y \sim N(2.4, 0.25)$, respectively.

May/June 2022

The independent random variables $X$ and $Y$ are distributed as $\text{Po}(2)$ and $\text{B}(20, \tfrac{1}{4})$ respectively.

May/June 2022

The masses, measured in kilograms, of large sacks of grain and small sacks of grain follow the distributions $\mathrm{N}(53,11)$ and $\mathrm{N}(14,3)$ respectively.

May/June 2022

A box of Seeds & Raisins has $S$ grams of seeds and $R$ grams of raisins. When the box is empty, its mass is $B$ grams. $S$, $R$ and $B$ are independent random variables, with $S \sim N(300, 45)$, $R \sim N(200, 25)$ and $B \sim N(15, 4)$. One complete box of Seeds & Raisins is picked at random.

May/June 2022

The random variables $X$ and $W$ are assigned probability density functions $f$ and $g$, given below: $f(x)=\begin{cases}p(a^2-x^2), & 0 \le x \le a, \\ 0, & \text{otherwise},\end{cases}$ $g(w)=\begin{cases}q(a^2-w^2), & -a \le w \le a, \\ 0, & \text{otherwise},\end{cases}$ where $a$, $p$ and $q$ are constants.

May/June 2022

Large rice packets are placed into cartons, with 20 packets selected at random for each carton. The packet masses are normally distributed with mean $1010\,\text{g}$ and standard deviation $3.4\,\text{g}$. The masses of empty cartons are independently normally distributed with mean $50\,\text{g}$ and standard deviation $2.0\,\text{g}$. Small packets of rice are packed in boxes. The total masses of full boxes are normally distributed with mean $6730\,\text{g}$ and standard deviation $15.0\,\text{g}$. The masses of the boxes and cartons are distributed independently of one another.

May/June 2023

The random variables $X$ and $Y$ are independent and have distributions $N(7, 3)$ and $N(6, 2)$ respectively. One value is chosen from each distribution. Find the probability that the absolute difference between the two chosen values is greater than $2$.

May/June 2023

The mass, in tonnes, of steel made each day at a factory is normally distributed with mean $65.2$ and standard deviation $3.6$. It may be assumed that the amount of steel produced on one day is independent of the amount produced on any other day. The factory earns $\$50$ profit for each tonne of steel produced.

May/June 2023

The random variable $X$ follows $N(31.2, 10.4^2)$. Two independent draws from $X$, labelled $X_1$ and $X_2$, are taken.

May/June 2024

The random variable $X$ follows the distribution $N(10, 12)$. Two values of $X$ are selected independently at random, and they are labelled $X_1$ and $X_2$.

May/June 2024

The kilogram masses of large and small bags of cement are independently distributed as $N(50,\,2.4)$ and $N(26,\,1.8)$, respectively.

May/June 2024

The distributions of the independent random variables $X$ and $Y$ are, respectively, $\text{Po}(1.9)$ and $\text{Po}(2.2)$.

May/June 2024

At an entertainment centre, the charge for playing a particular video game is $\$0.40$ for each minute. The number of minutes that people spend on the video game has mean $15$ and variance $9$.

May/June 2025

A student must be picked at random from a group of three. Explain how one fair throw of a six-sided dice could be used to decide which student is chosen.

May/June 2025

In Urberia, the masses of men, in kilograms, are distributed as $N(70.3, 5.9^2)$. A particular footbridge in Urberia has a maximum safe load of 1500 kg. If $n$ men are on the bridge, the chance that it is unsafe is below 0.01.

May/June 2025

Sanket takes part in a game with a biased die, where an even result is twice as likely as an odd result. The three even faces all have the same probability, and the three odd faces all have the same probability.

Oct/Nov 2010

A factory produces a large quantity of ropes whose lengths are either $3\text{ m}$ or $5\text{ m}$. The number of ropes of length $3\text{ m}$ is four times the number of ropes of length $5\text{ m}$.

Oct/Nov 2011

A fair tetrahedral die has four triangular faces, numbered $1$, $2$, $3$ and $4$. The result of a throw is the number shown on the face that comes to rest. The die is rolled three times. The random variable $X$ is the total of the three results.

Oct/Nov 2012

The discrete random variable $X$ is given this probability distribution: $x = -3, 0, 2, 4$ with probabilities $P(X = x) = p, q, r, 0.4$. It is also given that $\text{E}(X) = 2.3$ and $\mathrm{Var}(X) = 3.01$.

Oct/Nov 2012

James has a fair coin and a fair tetrahedral die with four faces labelled $1$, $2$, $3$, $4$. He tosses the coin one time and throws the die twice. The random variable $X$ is defined like this: if the coin lands heads, then $X$ is the total of the scores from the two die throws; if the coin lands tails, then $X$ is just the score from the first die throw.

Oct/Nov 2013

Rory owns $10$ cards. On four of them, a $3$ is shown, while the other six each show a $4$. He chooses three cards at random without replacement, then adds the numbers on those cards.

Oct/Nov 2013

Jodie flips a biased coin and rolls two fair tetrahedral dice. The probability that the coin lands on a head is $\frac{1}{3}$. Each die has four faces, labelled $1, 2, 3$ and $4$. Jodie’s score is worked out from the numbers on the faces where the dice come to rest.

Oct/Nov 2014

Four fair six-sided dice, with faces numbered $1, 2, 3, 4, 5, 6$, are rolled. Determine the probability that the four shown numbers add to $5$.

Oct/Nov 2014

Nadia is extremely forgetful. Whenever she signs in to her online bank, she has only a $40\%$ chance of typing her password correctly. On any one day, she is permitted $3$ unsuccessful attempts, after which the bank stops her from trying again until the next day.

Oct/Nov 2015

A fair spinner $A$ has edges labelled $1, 2, 3, 3$. A fair spinner $B$ has edges labelled $-3, -2, -1, 1$. Both spinners are spun once. Record the number on the edge where each spinner settles. Let $X$ denote the sum of the two numbers.

Oct/Nov 2015

A pair of fair six-sided dice numbered $1, 2, 3, 4, 5, 6$ are rolled, and both results are recorded. The difference between the two scores is defined in the following way. If the two scores are the same, the difference is zero. If the scores are different, the difference is the larger score minus the smaller score.

Oct/Nov 2016

Deeti has 3 red pens and 1 blue pen in her left pocket, and her right pocket also contains 3 red pens and 1 blue pen. Operation $T$ means that Deeti first takes one pen at random from her left pocket and puts it into her right pocket, and then takes one pen at random from her right pocket and puts it into her left pocket. The random variable $X$ gives the number of blue pens in Deeti’s left pocket after operation $T$ has been completed.

Oct/Nov 2016

This fair triangular spinner has sides labelled 1, 2 and 3. Each spin gives the number shown on the side it lands on. It is spun four times.

Oct/Nov 2016

Shown below is the probability distribution for the discrete random variable $X$: $x: 1,\;2,\;3,\;6$ $P(X = x): 0.15,\;p,\;0.4,\;q$

Oct/Nov 2017

A fair tetrahedral die has the faces labelled 1, 2, 3, 4. A biased coin has probability \(\frac{1}{3}\) of landing on a head when it is tossed. The die is thrown once and the value \(n\) on the face it lands on is recorded. The biased coin is then tossed \(n\) times. For instance, if the die lands on 3, the coin is tossed 3 times.

Oct/Nov 2017

A random variable $X$ is given by the probability distribution in the table below, with $p$ as a constant. The possible values are $x = -1, 0, 1, 2, 4$ and the corresponding probabilities are $P(X = -1) = p$, $P(X = 0) = p$, $P(X = 1) = 2p$, $P(X = 2) = 2p$, and $P(X = 4) = 0.1$.

Oct/Nov 2018

A fair red spinner has 4 sides, labelled $1, 2, 3, 4$. A fair blue spinner has 3 sides, labelled $1, 2, 3$. Each spinner’s score is the number on the side where it comes to rest. Both spinners are spun simultaneously. The random variable $X$ represents the score on the red spinner minus the score on the blue spinner.

Oct/Nov 2018

A fair 6-sided die has the numbers $-1, -1, 0, 0, 1, 2$ on its faces. A fair 3-sided spinner has edges numbered $-1, 0, 1$. After the die is thrown and the spinner is spun, the number showing on the top face of the die and the number on which the spinner settles are recorded. The total of these two numbers is represented by $X$.

Oct/Nov 2018

Benju rides to work every morning and has two routes from which to choose. He takes the hilly route with probability $0.4$ and the busy route with probability $0.6$. If he goes by the hilly route, the probability that he is late for work is $x$, and if he goes by the busy route, the probability that he is late for work is $2x$. The probability that Benju is late for work on any day is $0.36$.

Oct/Nov 2019

For a given species, the masses of female and male animals, measured in kilograms, follow the distributions $N(102, 27^2)$ and $N(170, 55^2)$ respectively.

Oct/Nov 2020

The diagram illustrates the probability density function, $f(x)$, for a random variable $X$. When $0 \leq x \leq a$, $f(x) = k$; for all other values, $f(x) = 0$.

Oct/Nov 2020

Before publication, a book of this kind is proofread and any mistakes are corrected. For costing, each error is put into one of two classes: minor or major. The numbers of minor and major errors in a book are represented by the independent distributions $N(380, 140)$ and $N(210, 80)$ respectively. Assume that no continuity corrections are required when these models are used. One book of this type is selected at random.

Oct/Nov 2020

The masses, measured in kilograms, of female and male animals of a particular species follow the distributions $N(102, 27^2)$ and $N(170, 55^2)$ respectively.

Oct/Nov 2020

Let the random variable $M$ represent the mass, in kilograms, of a block of cheese sold in a supermarket. A random sample of 40 blocks has these summary statistics: $n = 40$, $\sum m = 20.50$, $\sum m^2 = 10.7280$.

Oct/Nov 2021

In each month, the company sells $X$ kg of brown sugar and $Y$ kg of white sugar, with $X$ and $Y$ having the independent distributions $N(2500, 120^2)$ and $N(3700, 130^2)$ respectively.

Oct/Nov 2022

The masses, measured in grams, of small and large bags of flour follow the distributions $N(510, 100)$ and $N(1015, 324)$ respectively. André chooses $4$ small bags of flour and $2$ large bags of flour at random.

Oct/Nov 2022

In each month, the firm sells $X$ kg of brown sugar and $Y$ kg of white sugar, with $X$ and $Y$ independent and distributed as $N(2500, 120^2)$ and $N(3700, 130^2)$ respectively.

Oct/Nov 2022

The daily masses, measured in kilograms, of chemicals $A$ and $B$ made by a factory are described by the independent random variables $X$ and $Y$ respectively, where $X \sim N(10.3,\,5.76)$ and $Y \sim N(11.4,\,9.61)$. The revenue from the chemicals is $\$2.50$ for each kilogram of $A$ and $\$3.25$ for each kilogram of $B$.

Oct/Nov 2023

A factory’s daily masses, in kilograms, of chemicals $A$ and $B$ are represented by the independent random variables $X$ and $Y$ respectively, where $X \sim N(10.3,\ 5.76)$ and $Y \sim N(11.4,\ 9.61)$. The revenue from the chemicals is $\$2.50$ per kilogram for $A$ and $\$3.25$ per kilogram for $B$.

Oct/Nov 2023

The masses, measured in kilograms, of small bags of wheat and large bags of wheat are independently distributed as $N(16.0, 0.4)$ and $N(51.0, 0.9)$, respectively.

Oct/Nov 2024

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

Oct/Nov 2024

The random variables $X$ and $Y$ are independent, with $X \sim \mathrm{Po}(3)$ and $Y \sim \mathrm{Po}(2)$ respectively.

Oct/Nov 2025

The masses, measured in kilograms, of the large bags and the small bags of potatoes are independently distributed as $N(2.5, 0.05)$ and $N(0.8, 0.02)$ respectively.

Oct/Nov 2025

The random variable $X$ follows a normal distribution with mean 10 and standard deviation 3, while the independent random variable $Y$ follows a Poisson distribution with mean 4.

Oct/Nov 2025

The masses, measured in kilograms, of large and small bags of potatoes follow the independent distributions $N(2.5, 0.05)$ and $N(0.8, 0.02)$ respectively.

Oct/Nov 2025

$L$ and $S$ denote the masses, in kg, of large and small bags of tomatoes, with distributions $L \sim N(2.10, 0.12)$ and $S \sim N(1.51, 0.09)$ respectively. Tomatoes cost $\$4.30$ per kg.

Oct/Nov 2025