Continuous random variables

A fair eight-sided die is numbered $1, 2, 3, 4, 5, 6, 7, 8$ on its faces. When the die is rolled, the score is the number showing on the face it lands on. The die is rolled twice. Event $R$ is ‘one score is exactly $3$ greater than the other score’. Event $S$ is ‘the product of the scores is more than $19$’.

Feb/March 2016

The lengths of time a garage needs to attach a tow bar to a car are normally distributed with mean $m$ hours and standard deviation $0.35$ hours. It is given that $95\%$ of the times exceed $0.9$ hours.

Feb/March 2016

The weights, measured in kilograms, of cereal packets were recorded to $4$ significant figures. The stem-and-leaf diagram below displays the data. Key: $748\;|\;5$ indicates $0.7485\text{ kg}$.

Feb/March 2017

The middle-finger lengths, measured in centimetres, of women in Raneland follow a normal distribution with mean $\mu$ and standard deviation $\sigma$. It is given that $25\%$ of these women have fingers longer than $8.8\text{ cm}$, while $17.5\%$ have fingers shorter than $7.7\text{ cm}$.

Feb/March 2017

The packet weights for one particular type of biscuit are assumed to be normally distributed, with mean 400 grams and standard deviation $\sigma$ grams.

Feb/March 2018

Train journey times, measured in minutes, between Alphaton and Beeton follow a normal distribution with mean $140$ and standard deviation $12$.

Feb/March 2019

Bottles of Lanta each hold about $300$ ml of juice. If the juice volume, in millilitres, in a bottle is $300 + X$, then $X$ is a random variable with probability density function defined by $f(x) = \begin{cases} \frac{3}{4000}(100 - x^2), & -10 \leq x \leq 10, \\ 0, & \text{otherwise}. \end{cases}$

Feb/March 2020

The diagram displays the graph of the probability density function, $f$, for the random variable $X$.

Feb/March 2021

A ball is released down a slope in a game and then travels along a track until it comes to rest. Let the distance, in metres, covered by the ball be represented by the random variable $X$ with probability density function $f(x) = \begin{cases} -k(x - 1)(x - 3), & 1 \le x \le 3, \\ 0, & \text{otherwise}. \end{cases}$ where $k$ is a constant.

Feb/March 2022

The diagram displays the graph of the probability density function, $f$, for a random variable $X$ that can take only values from $x = 0$ to $x = 3$. The graph is symmetric about the line $x = 1.5$.

Feb/March 2023

The probability density function graph $f$ for a random variable $X$ is symmetric about the line $x = 2$. It is stated that $\mathrm{P}(2 < X < 5) = \frac{117}{256}$.

Feb/March 2024

The diagram is the graph of the probability density function, $f$, for a random variable $X$. It is a straight line joining $(0, a)$ to $(2, b)$, with $a$ and $b$ both positive constants. For all other values of $x$, $f(x) = 0$.

Feb/March 2025

The table below displays the probability distribution of the discrete random variable $X$.

May/June 2010

The random variable $X$ denotes the time, measured in minutes, that Jannon needs to repair a bicycle puncture. $X$ has a normal distribution with mean $\mu$ and variance $\sigma^2$.

May/June 2010

The lengths of new pencils follow a normal distribution with mean $11\text{ cm}$ and standard deviation $0.095\text{ cm}$.

May/June 2010

The random variable $X$ has a normal distribution with mean $\mu$ and standard deviation $\sigma$.

May/June 2010

For children of this age, jump heights are normally distributed. On average, $8$ children out of $10$ can jump higher than $127\text{ cm}$, and $1$ child out of $3$ can jump higher than $135\text{ cm}$.

May/June 2010

The random variable $X$ may take the 8 integer values in the set $\{-2, -1, 0, 1, 2, 3, 4, 5\}$. The chance that $X$ equals $0$ is $\frac{1}{10}$. Every other value of $X$ has the same probability.

May/June 2011

The random variable $X$ has a normal distribution with mean $\mu$ and standard deviation $\sigma$. You are told that $3\mu = 7\sigma^2$ and that $P(X > 2\mu) = 0.1016$. Find $\mu$ and $\sigma$.

May/June 2011

For Scotland in November, the average proportion of cloudy days is $80\%$. Assume that the weather on one day is independent of the weather on every other day.

May/June 2011

The lengths, measured in centimetres, of drinking straws made in a factory follow a normal distribution with mean $\mu$ and variance $0.64$. It is stated that $10\%$ of the straws are less than $20\,\text{cm}$ long.

May/June 2011

It is stated that $X \sim N(28.3,\,4.5)$.

May/June 2012

The durations needed to perform Beethoven’s Sixth Symphony may be regarded as following a normal distribution with mean $41.1$ minutes and standard deviation $3.4$ minutes. Three performances of this symphony are selected at random.

May/June 2012

The lengths, in cm, of trout in a fish farm follow a normal distribution. 96% of the lengths are below 34.1 cm, and 70% of the lengths are above 26.7 cm.

May/June 2012

The random variable $Y$ follows a normal distribution, and its mean is five times its standard deviation.

May/June 2013

Each can of lemon juice should contain $440\,\text{ml}$ of juice. The actual amount of juice in a can is stated to be normally distributed with mean $445\,\text{ml}$ and standard deviation $3.6\,\text{ml}$.

May/June 2013

Buildings in one city-centre area are grouped by height into tall, medium or short. Their heights may be represented by a normal distribution with mean $50$ metres and standard deviation $16$ metres. Any building taller than $70$ metres is put in the tall category.

May/June 2013

The petrol consumption for one particular type of car follows a normal distribution with mean 24 kilometres per litre and standard deviation 4.7 kilometres per litre.

May/June 2014

The lengths of a particular variety of white radish follow a normal distribution with mean $\mu$ cm and standard deviation $\sigma$ cm. $4\%$ of the radishes have lengths greater than $12$ cm, while $32\%$ exceed $9$ cm.

May/June 2014

The chance that Wenjie goes out with her friends on a given day is $\frac{1}{7}$. 252 days are selected at random.

May/June 2014

The length of each phone call Moses makes has a normal distribution with mean 6.5 minutes and standard deviation 1.76 minutes.

May/June 2014

Car lengths in this city, measured in metres, are normally distributed with mean $\mu$ and standard deviation $0.714$.

May/June 2015

Zak goes for a run once each week. The length of time he takes, measured in minutes, follows a normal distribution with mean $35.2$ and standard deviation $4.7$. Determine the expected number of days in a year ($52$ weeks) when Zak’s run takes less than $30$ minutes.

May/June 2015

The heights of books in a library, measured in cm, follow a normal distribution with mean $21.7$ and standard deviation $6.5$. Any book taller than $29$ cm is described as ‘large’.

May/June 2015

Maize plant heights in Mpapwa follow a normal distribution with mean $1.62\,\text{m}$ and standard deviation $\sigma\,\text{m}$. The chance that a randomly selected plant is taller than $1.8\,\text{m}$ is $0.15$.

May/June 2016

Peter’s journey to the shop plus the purchase of a newspaper takes a time, measured in minutes, that is normally distributed with mean 9.5 and standard deviation 1.3.

May/June 2016

The heights of school desks follow a normal distribution with mean $69\,\text{cm}$ and standard deviation $\sigma\,\text{cm}$. It is given that $15.5\%$ of the desks are taller than $70\,\text{cm}$.

May/June 2016

Kadijat recorded the masses, $x$ grams, of 30 chocolate buns. Her findings are given by $\sum (x-k) = 315$, $\sum (x-k)^2 = 4022$, where $k$ is a constant. The buns have a mean mass of $50.5$ grams.

May/June 2017

Let $X$ be normally distributed with mean $\mu$ and standard deviation $\sigma$. It is known that $\sigma = 0.25\mu$ and that $P(X < 6.8) = 0.75$.

May/June 2017

Video lengths for a particular popular song are normally distributed, with mean $3.9$ minutes. $18\%$ of the videos run for more than $4.2$ minutes.

May/June 2017

On any day, the chance that George goes swimming is $\frac{1}{3}$.

May/June 2017

The random variable $X$ is distributed as $N(\mu, \sigma^2)$, with $\mu = 1.5\sigma$. A value of $X$ is selected at random. Find the probability that this selected value of $X$ is greater than $0$.

May/June 2017

The distance that tyres of a certain make can cover before replacement is needed has a normal distribution. A large survey of these tyres showed that the probability of the distance exceeding $36\,800\text{ km}$ is $0.0082$, while the probability of it exceeding $31\,000\text{ km}$ is $0.6915$. Determine the mean and standard deviation of the distribution.

May/June 2018

In Pelmerdon, $22\%$ of families have a dishwasher.

May/June 2018

The amount of soup in Super Soup cartons follows a normal distribution with mean $\mu$ millilitres and standard deviation $9$ millilitres. Testing has shown that $10\%$ of cartons contain under $440$ millilitres of soup. Find the value of $\mu$.

May/June 2018

The durations, $t$ minutes, of $242$ telephone calls made by one family over a span of $1$ week are set out in the frequency table below.

May/June 2018

Let $X$ be distributed as $N(-3,\sigma^2)$. The chance that a value of $X$ selected at random is greater than 0 is $0.25$.

May/June 2018

The diameters of apples in an orchard are normally distributed, with mean $5.7\text{ cm}$ and standard deviation $0.8\text{ cm}$. Apples whose diameters lie between $4.1\text{ cm}$ and $5.0\text{ cm}$ may be used for toffee apples.

May/June 2018

Adult female giraffe weights follow a normal distribution with mean $830\text{ kg}$ and standard deviation $120\text{ kg}$.

May/June 2019

The amount of ink in one kind of ink cartridge has a normal distribution with mean $30\text{ ml}$ and standard deviation $1.5\text{ ml}$. Staff in an office use $8$ of these cartridges each month.

May/June 2019

The crossing time, measured in minutes, for a ferry to travel across a lake follows a normal distribution with mean $85$ and standard deviation $6.8$.

May/June 2019

The waiting time, $T$ minutes, for a passenger at a particular bus stop is represented by the probability density function $f(t) = \begin{cases} \frac{3}{4000}(20t - t^2), & 0 \leq t \leq 20, \\ 0, & \text{otherwise.} \end{cases}$

May/June 2020

A random variable $X$ has probability density function defined by $$f(x) = \begin{cases} \frac{k}{x^2}, & 1 \le x \le a, \\ 0, & \text{otherwise}, \end{cases}$$ where $k$ and $a$ are positive constants.

May/June 2020

The random variable $X$, measured in centimetres and representing the length of worms of a certain type, is modelled by the probability density function $f(x) = \begin{cases} \frac{6}{125}(10 - x)(x - 5), & 5 \le x \le 10, \\ 0, & \text{otherwise}. \end{cases}$

May/June 2020

The probability density function, $f$, for the random variable $X$ is defined as $f(x) = \begin{cases} k(6x - x^2), & 0 \leq x \leq 6, \\ 0, & \text{otherwise}, \end{cases}$ with $k$ as a constant.

May/June 2021

The random variable $X$ may take any value in the interval $1 \leq x \leq p$, with $p$ fixed. The diagram presents the graph of the probability density function of $X$.

May/June 2021

Alethia represents the delay time, in minutes, of her train on any day by the random variable $X$, with probability density function $f(x) = \begin{cases} \frac{3}{8000}(x - 20)^2, & 0 \le x \le 20, \\ 0, & \text{otherwise}. \end{cases}$

May/June 2021

A random variable $X$ has probability density function $f$. The plot of $f(x)$ is a horizontal line segment, parallel to the $x$-axis, running from $x = 0$ to $x = a$, where $a$ is a positive constant.

May/June 2022

Let the random variable $X$ have probability density function given by $$f(x)=\begin{cases}\frac{3}{16}(4x-x^2), & 2 \leq x \leq 4, \\ 0, & \text{otherwise}.\end{cases}$$

May/June 2022

The graph of the function $f$ is a line segment running from $(0, 0)$ to $(2, 1)$. Show that $f$ could be a probability density function.

May/June 2023

The diagram gives the graph of the probability density function, $f$, for a random variable $X$ whose possible values lie only in the interval from $0$ to $4$. Over this interval the graph is a straight line. Show that $f(x) = kx$ for $0 \le x \le 4$, where $k$ is a constant to be found.

May/June 2023

The random variable $X$ is given the probability density function $f$, with $f(x) = \begin{cases} \frac{3}{2}(1 - x^2), & 0 \leq x \leq 1, \\ 0, & \text{otherwise}. \end{cases}$

May/June 2023

The random variable $X$ has probability density function $f$. Its graph, $y = f(x)$, is a semicircle centred at $(0,0)$ with radius $\sqrt{\tfrac{2}{\pi}}$, and it lies entirely above the $x$-axis. For all other values, $f(x) = 0$ (see diagram).

May/June 2023

The diagram depicts the probability density function, $f$, for a random variable $X$. It is a quarter circle that lies wholly in the first quadrant, has centre $(0,0)$ and radius $a$, where $a$ is a positive constant. For all other $x$, $f(x)=0$.

May/June 2024

A student wants to estimate the proportion, $p$, of students at her college who have exactly one brother. She takes a random sample of 50 students at her college and discovers that 18 of them have exactly one brother. She then works out an approximate $\alpha\%$ confidence interval for $p$ and finds that the lower boundary of the interval is $0.244$ correct to 3 significant figures.

May/June 2024

The probability density function, $f$, for the random variable $X$ is defined by $f(x)=\begin{cases}k(1+\cos x), & 0 \le x \le \pi,\\ 0, & \text{otherwise},\end{cases}$ where $k$ denotes a constant.

May/June 2024

The random variable $X$ is described by the probability density function $f$, where $f(x) = \begin{cases} ax - x^3, & 0 \leq x \leq \sqrt{2}, \\ 0, & \text{otherwise}, \end{cases}$ and $a$ is a constant.

May/June 2024

$X$ is a random variable with probability density function defined as $f(x) = \begin{cases} 1 + \cos \pi x, & 0 \leq x \leq 1, \\ 0, & \text{otherwise}. \end{cases}$

May/June 2025

The probability density function of the random variable $X$ is given by $f(x) = \begin{cases} \dfrac{kx^2}{a^2}, & 0 \leq x \leq a, \\ 0, & \text{otherwise}, \end{cases}$ where $k$ and $a$ are positive constants.

May/June 2025

Let $X$ denote a random variable whose probability density function is defined by $f(x)=\begin{cases} ax, & 0 \leq x \leq b, \\ 0, & \text{otherwise}, \end{cases}$ with $a$ and $b$ as constants.

May/June 2025

The diagram illustrates the probability density function for a random variable $X$. For $x = 0$ to $x = a$, the graph is a straight line passing through $O$ with gradient $k$, where $k$ and $a$ are positive constants. For all other values of $x$, $f(x) = 0$. The median of $X$ is given as $\sqrt{2}$.

May/June 2025

The times that students take to get out of bed in the morning can be represented by a normal distribution with mean 26.4 minutes and standard deviation 3.7 minutes.

Oct/Nov 2010

The distance that the Zotoc car can cover on $20$ litres of fuel follows a normal distribution with mean $320$ km and standard deviation $21.6$ km. The distance that the Ganmor car can cover on $20$ litres of fuel follows a normal distribution with mean $350$ km and standard deviation $7.5$ km. Each car is given $20$ litres of fuel and driven towards a destination $367$ km away.

Oct/Nov 2010

State the distribution name and propose appropriate numerical parameters that could be used to model the weights in kilograms of female 18-year-old students.

Oct/Nov 2010

The lengths of time that people spend at a particular dentist are mutually independent and follow a normal distribution with mean $8.2$ minutes. $79\%$ of the people who visit this dentist have appointments lasting under $10$ minutes.

Oct/Nov 2010

For a butternut squash seed that is sown, the chance that it germinates is $0.86$, independently of all the other seeds. A market gardener sows $250$ of these seeds.

Oct/Nov 2011

The letter weights sent by one business follow a normal distribution with mean $20\,\mathrm{g}$. It is given that $94\%$ of the letters have weights lying within $12\,\mathrm{g}$ of the mean.

Oct/Nov 2011

In one country, winter daily minimum temperature, measured in $^\circ\text{C}$, follows the distribution $N(8, 24)$. Find the probability that a winter day chosen at random in this country has a minimum temperature from $7^\circ\text{C}$ to $12^\circ\text{C}$. In another country, the winter daily minimum temperature, in $^\circ\text{C}$, has a normal distribution with mean $\mu$ and standard deviation $2\mu$.

Oct/Nov 2011

Suppose the random variable $X$ has a normal distribution, with mean $\mu$ equal to three times the standard deviation $\sigma$. It is also given that $\mathrm{P}(X < 25) = 0.648$.

Oct/Nov 2011

Roll lengths of parcel tape are normally distributed with mean $75\,\text{m}$, and $15\%$ of the rolls are shorter than $73\,\text{m}$. Alison purchases 8 rolls of parcel tape.

Oct/Nov 2012

The random variable $X$ represents the company's daily profit, measured in thousands of dollars. It has a normal distribution with mean $6.4$ and standard deviation $5.2$.

Oct/Nov 2012

For a normal distribution with mean $9.3$, the chance that a value chosen at random is above $5.6$ is $0.85$.

Oct/Nov 2012

The random variable $X$ has distribution $X \sim N(82, 126)$.

Oct/Nov 2012

The information provided is that $X \sim N(30, 49)$, $Y \sim N(30, 16)$ and $Z \sim N(50, 16)$.

Oct/Nov 2013

Carrot lengths of this type follow a normal distribution with mean $14.2$ cm and standard deviation $3.6$ cm.

Oct/Nov 2013

It is known that $X \sim N(1.5, 3.2^2)$.

Oct/Nov 2013

For a packet from a particular brand of cereal, the fibre content is normally distributed with mean $160$ grams. $19\%$ of packets of cereal contain more than $190$ grams of fibre.

Oct/Nov 2013

For each of the 55 students, the distance from home to college, rounded to the nearest kilometre, was noted. These distances are presented in the table below. Dominic is required to draw a histogram to show the data. Dominic’s diagram appears below.

Oct/Nov 2013

A factory manufactures flower pots. The diameters of their bases are normally distributed with mean $14\text{ cm}$ and standard deviation $0.52\text{ cm}$.

Oct/Nov 2013

The random variable $X$ has a normal distribution with mean $82$ and standard deviation $7.4$. Determine the value of $q$ for which $P(82-q < X < 82+q) = 0.44$.

Oct/Nov 2013

A farmer observes that the weights of sheep on his farm follow a normal distribution with mean $66.4\,\text{kg}$ and standard deviation $5.6\,\text{kg}$.

Oct/Nov 2014

The variable $X$ hours, representing the amount of sleep people get in one night, follows a normal distribution with mean $7.15$ hours and standard deviation $0.88$ hours. Find the probability that a person selected at random sleeps for under $8$ hours in one night.

Oct/Nov 2014

Tea packets are labelled as containing 250 g. The true mass of tea in a packet follows a normal distribution with mean 260 g and standard deviation $\sigma$ g. Any packet whose mass is below 250 g is described as ‘underweight’.

Oct/Nov 2014

Gem stones drawn from one mine have weights, $X$ grams, and these are normally distributed with mean $1.9\text{ g}$ and standard deviation $0.55\text{ g}$. For sale, the gem stones are split into three weight bands: Small: below $1.2\text{ g}$; Medium: from $1.2\text{ g}$ to $2.5\text{ g}$; Large: above $2.5\text{ g}$.

Oct/Nov 2014

The random variable $X$ follows the distribution $N(\mu, \sigma^2)$. It is known that $P(X < 54.1) = 0.5$ and $P(X > 50.9) = 0.8665$.

Oct/Nov 2015

Robert works part-time delivering newspapers. On several days, he recorded the time taken to complete his job, correct to the nearest minute. He used his results to complete the table below; the two missing entries are shown as $a$ and $b$.

Oct/Nov 2015

A biased die has faces numbered $1, 2, 3, 4, 5$ and $6$. The probabilities of obtaining odd numbers are all equal. The probabilities of obtaining even numbers are all equal. The chance of an odd number is twice the chance of an even number.

Oct/Nov 2015

Daily sales, measured in litres, at a petrol station are normally distributed with mean $4520$ and standard deviation $560$. Determine for how many of the $365$ days in the year the daily sales are expected to exceed $3900$ litres.

Oct/Nov 2015

Over 250 working days, the time taken, $t$ hours, to deliver letters along a particular route is recorded each day. The average time is $2.8$ hours. If $\sum (t - 2.5)^2 = 96.1$, determine the standard deviation of the times taken.

Oct/Nov 2015

Under these conditions, the germination time of cucumber seeds follows a normal distribution with mean $125$ hours and standard deviation $\sigma$ hours.

Oct/Nov 2015