Linear Combinations of Random Variables (Cambridge International A-Level Mathematics): Exam Questions

The discrete random variable $X$ has the probability distribution shown in the following table.

(i) Show that $E(X)=2.2$.

(ii) Show that $Var(X)=1.56$.

Complete the row in the table below to show the probability distribution for $X+5$.

Use the second and third rows in table in part (b) with the formula $E(X+5)=\sum(x+5)p$ to verify that $E(X+5)=E(X)+5$.

Use the second and third rows in table in part (b) with the formulae $E(X+5)^{2}=\sum(x+5)^{2}p$ and $Var(X+5)=E(X+5)^{2}-(E(X+5))^{2}$ to verify that $Var(X+5)=Var(X)$.

The discrete random variable $X$ has the probability distribution shown in the following table.

(i) Show that $E(X)=1.2$

(ii) Show that $Var(X)=1.76$

Complete the row in the table below to show the probability distribution for $3X$.

Use the second and third rows in table in part (a) with the formula $E(3X)=\sum(3x)p$ to verify that $E(3X)=3E(X)$.

Use the second and third rows in table in part (a) with the formula $E(3X)^{2}=\sum(3x)^{2}p\mathrm{and}Var(3X)=E(3X)^{2}-(E(3X))^{2}$ to verify that $Var(3X)=9Var(X)$.

$X$ is a random variable such that $E(X)=5\mathrm{and}Var(X)=3$.

Using the formulae $E(aX+b)=aE(X)+b\mathrm{and}Var(aX+b)=a^{2}Var(X)$, find the mean and variance of the following random variables:

(i) $4X+1$

(ii) $7X-2$

(iii) $5-X$

If $X$ is normally distributed, then $aX+b$ is normally distributed for any constants $a$ and $b$.

Using the formulae $E(aX+b)=aE(X)+b\mathrm{and}Var(aX+b)=a^{2}Var(X)$, show that, if $X \sim N ( \\mu , \\sigma ^{2} )$ then:

$a X + b \sim N ( a \\mu + b , a^{2} \\sigma ^{2} )$

The random variable $X \sim N ( 50 , 16 )$, write down the distribution for the random variable:

(i) $2X+10$

(ii) $3X-20$

(iii) $50-X$

The random variable $Y \sim N ( 15 , 5^{2} )$

(i) Find $P(Y<18)$

(ii) Write down the distribution of $4Y+40$

(iii) Find $P(4Y+40<112)$

If $X$ and $Y$ are two independent random variables, then:

$E(X\pm Y)=E(X)\pm E(Y)\mathrm{and}Var(X\pm Y)=Var(X)+Var(Y)$

The random variable $S$ has a mean of 10 and a variance of 5 and the random variable $T$ has a mean of 15 and a variance of 7.

Find the mean and variance of:

(i) $S+T$

(ii) $S-T$

Using the above formulae and the formulae $E(aX+b)=aE(X)+b\mathrm{and}Var(aX+b)=a^{2}Var(X)$, it follows that if $X$ and $Y$ are two independent random variables then:

$E(aX+bY)=aE(X)+bE(Y)andVar(aX+bY)=a^{2}Var(X)+b^{2}Var(Y)$

Find the mean and variance of $3S+2T$.

If $X\mathrm{and}Y$ are normally distributed and independent then $X+Y$ is also normally distributed. As $aX+b$ is also normally distributed it follows that $aX+bY$ is normally distributed.

In particular, if X tilde N ( mu _ x , sigma _ x ^ 2 ) and Y tilde N ( mu _ y , sigma _ y ^ 2 ) are two independent distributions, then:

$X + Y \sim N ( a \mu_{x} + b \mu_{y} , a^{2} \sigma_{x}^{2} + b^{2} \sigma_{y}^{2} )$

The random variables R tilde N ( 10 , 16 ) , S tilde N ( 15 , 9 ) and T tilde N ( 25 , 25 ) are independent.

Write down the distribution of:

(i) $R+S$

(ii) $2S+3T$

The random variable is normally distributed with a mean of 10 and a standard deviation of 3.

State the distribution of the sum of three independent observations of the random variable $X$.

State the distribution of the random variable $3X$.

The random variable $X$ has a mean of 40 and a variance of 36.

Find the mean and variance of:

(i) $2X+5$

(ii) $3X-10$

(iii) $100-2X$

It is known that $X$ follows a Normal distribution.

Find $P(2X+1<75)$.

The random variable $X$ has mean $μ$ and variance $σ^{2}$. Given that

$E(2X+7)=16\,Var(2X+7)=5$

Find the value of $μ$ and the value of $σ^{2}$.

The random variable $X$ has mean and variance of 4, and the random variable $Y$ has mean and variance of 5. $X\mathrm{and}Y$ are independent.

Find the mean and variance of the random variable:

(i) $X+Y$

(ii) $3X+2Y$

(iii) $X-Y$

(iv) $3X-2Y$

Given that $X\mathrm{and}Y$ follow Normal distributions, find:

(i) $P(X+Y<4)$

(ii) $P(X-Y>4)$

A game involves flipping a biased coin where the probability of landing on tails is 0.2, otherwise the coin lands on heads. In the game the coin is flipped twice and the number of tails, $T$, is recorded.

Draw up the probability distribution table for $T$.

To calculate the score of the game, a player multiplies the number of tails by 10 and then subtracts 5.

Find the mean score.

Find the standard deviation of the scores.

The probability distributions for two independent random variables $S\mathrm{and}T$ are shown below.

Find the value of $E(S)$.

Find the value of $E(16S+15T)$.

Given that $Var(T)=11.5$, find the value of $Var(S+2T)$.

The distributions of the independent random variables $X\mathrm{and}Y$ are $N(8,2)\mathrm{and}N(9,3)$ respectively.

Find the probability that

(i) $2X+3Y>44$

(ii) The sum of two observations from $X$ and three observations from $Y$ are greater than 44.

Explain why your answers to (a) part (i) and (ii) are different.

Malik is fighting a boss in a video game. The loading times for the boss fight, $L$, follow a Normal distribution with mean of 14.1 seconds and variance of 6.1 seconds². The times it takes Malik to beat the boss, $B$, follow a Normal distribution with mean of 101.7 seconds and variance of 243.5 seconds².

The time taken to complete the level, $C$, is the sum of the loading time and the time taken to beat the boss.

State an assumption that is needed to model $C$ using a Normal distribution.

Assuming the assumption in part (a) is true, write down the distribution for $C$.

Find the probability that it takes Malik between 100 and 120 seconds to complete the level.

The random variable $X$ has mean 8 and variance 15. Given that

$E(aX+b)=23\,Var(aX+b)=135$

Find the two possible values of $b$.

The probability distribution for the random variable $X$ is shown below.

Find the value of:

(i) $E(\frac{X+7}{2})$

(ii) $Var(\frac{X}{5}-2)$

(iii) $Var(100-2X)$

(iv) $E(3X^{2}-1)$

Show that $E(X-E(X))=0$

The random variable $X$ has mean of $μ$ and standard deviation of $σ$. The random variable $Y$ has mean of 3 and standard deviation of 4. Given that

$E(2X+5Y)=25\,Var(2X+5Y)=724$

Find the value of $μ$ and the value of $σ$.

State the assumption that has been made about the random variables $X\mathrm{and}Y$.

A fair six-sided dice labelled with numbers 1, 1, 2, 2, 3, 3 is rolled and the number it lands on is denoted $D$.

Show that $E(D)=2$ and find $Var(D)$.

The dice is rolled twice and the arithmetic mean of the two numbers, which is denoted ̅D is calculated.

Draw up a probability distribution table for ̅D.

Find E(̅D) and Var(̅D).

Frank has a variable tariff for his electricity and gas bills. His monthly electricity bill is $E and his monthly gas bill is $G. $E\mathrm{and}G$ are independent random variables with distributions $N(85,9.4^{2})\mathrm{and}N(53,12.45)$ respectively.

Find the probability that the total electricity and gas bill in a month exceeds $150.

Frank has a part-time job tutoring college students. His monthly income from this job can be modelled as a Normal distribution with mean $504 and standard deviation $41. Frank uses this income to pay for his gas and electricity bills, he puts the remaining money into his partner’s bank account each month.

(i) Find the probability that Frank puts between $350 and $450 into his partner’s bank account in a month.

(ii) What assumption did you make in part (b)(i) regarding his income and his bills.

Veronica, a taxi driver in London, charges her customers a fixed fee of £5 plus £1.20 per mile. The lengths of her customers’ journeys are normally distributed with mean 16.7 miles and standard deviation 4.1 miles.

Find the standard deviation of the prices of Veronica’s taxi rides.

Find the probability that a taxi ride will cost less than £30.

Find the probability that the total cost of two independent taxi rides is more than £60.

On a bank holiday, Veronica doubles her prices.

Find the variance of the prices of Veronica’s taxi ride on a bank holiday.

Find the probability that a taxi ride on a bank holiday will cost more than £60.

The random variable $X$ has the distribution $N(5.9,2.1^{2})$.

Find the probability that the sum of 50 independent observations of exceeds 300.

Hence find the probability that the arithmetic mean of 50 independent observations of $X$ is less than 6.

Dinah’s Diner is famous for its triple burger which is made up of three beef patties, two rashers of bacon and a toasted bread bun. The mass, in grams, of a beef patty follows the distribution $N(110,6^{2})$. The mass, in grams, of a rasher of bacon follows the distribution $N(30,5^{2})$. The mass, in grams, of a toasted bread bun follows the distribution $N(50,3^{2})$.

Estimate the proportion of triple burgers at Dinah’s Diner that have a mass of more than 450 g.

State, with a reason, whether the probability that the total mass of two triple burgers exceeding 900 g is equal to your answer in part (a).

The random variable $X$ has mean 2 and variance 3 and the random variable $Y$ has mean 5 and variance 6. Given that $X\mathrm{and}Y$ are independent and that

$E(aX+bY)=-4\,Var(aX+bY)=51$

Find the integer values of $a$ and $b$.

The random variable $X$ has mean $μ$ and variance $σ^{2}$. The arithmetic mean of $n$ independent observations of $X$ is denoted by ̅X.

Show that E(̅X)=μ and Var(̅X)=\frac{σ^{2}}{n}.

40 independent observations are taken from the distribution $N(30,5^{2})$ and the arithmetic mean, ̅Y, is calculated.

(i) State, with a reason, whether the Central Limit Theorem is needed to model ̅Y as a Normal distribution.

(ii) Find $P(29<Y<30)$.

The probability of rolling a 6 on a biased die is $p$. In a game, this die is rolled twice and then the player multiplies the number of times the die lands on a 6 by 50 and then adds 5 to calculate the score.

(i) Given that the mean score is 15, draw up a probability distribution table for the number of times the die lands on a 6 when rolled twice.

(ii) Find the variance of the scores.

Two friends, Forrest and Gumpy, are planning to run a marathon together. The distributions F tilde N ( 253 , 95 ) and G tilde N ( 281 , 52 ) are used to model the times in minutes it takes Forrest and Gumpy to complete a marathon respectively. It can be assumed that their times are independent.

Find the probability that Forrest completes the marathon quicker than Gumpy.

Find the probability that Gumpy is still running the marathon one hour after Forrest has completed it.

Find the probability that their times taken to complete the marathon differ by more than 5 minutes.

A paper bag can hold 20 kg before it breaks. The mass of an orange is modelled using a Normal distribution with mean 260 g and standard deviation 12 g. The mass of each orange is independent from the others.

Find the maximum number of oranges that a paper bag can hold before the probability of the bag breaking exceeds 0.1%.

Roger is considering buying a new pet. He has researched the prices, in €, of rabbits, chinchillas and degus. The information is shown in the table below. The prices of the three types of animals are normally distributed and independent of each other.

Find the probability that the cost of two independently bought degus is less than €70.

Find the probability that a randomly selected degu is more expensive than a randomly selected rabbit.

Find the probability that a randomly selected chinchilla is more than five times as expensive as a randomly selected rabbit.

Roger and his housemate Lucy have decided to buy one of each type of pet for their house. Roger loves rabbits so he will pay for the rabbit himself, he will pay 50% of the cost for the chinchilla and 10% of the cost for the degu.

Find the probability that, in total, Roger pays less than €100 for the three pets.

The random variables $X \sim N ( 50 , 9^{2} ) a n d Y \sim N ( 400 , 150 )$ are independent.

Find $P(Y<7X+40)$.

There’s a 99.95% chance that the sum of a random observation of $X$and a random observation of $Y$ is bigger than $k$. Find the value of $k$.

Find the probability that the sum of three independent observations of $X$ is more than one third of one observation of $Y$.

In a video game a player gets points for completing a level and for defeating enemies, these points are independent of each other. The amount of points a player gets for completing the level and for defeating an enemy can be modelled as L tilde N ( 500 , 220 ) and E tilde N ( 150 , 85 ) respectively.

In a bonus stage, the points for completing the level are tripled and there are five enemies (points for defeating enemies are not tripled), the total score is the sum of the points for completing the level and defeating the enemies. The top 10% of scores make the leadership board.

Estimate the minimum score that would make the leadership board.