Linear Combinations of Random Variables (Cambridge (CIE) A Level Maths: Probability & Statistics 2): Exam Questions

The random variable $X$ has mean $\mu$ and variance ${\sigma }^2$. It is given that

$\text{E}(2X+7)=16$

$\text{Var}(2X+7)=5$

Find the value of $\mu$ and the value of ${\sigma }^2$.

The random variable $X$ has mean and variance of 4, and the random variable $Y$ has mean and variance of 5. $X$ 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 $X$ and $Y$ follow normal distributions, find:

(i) $\text{P}(X+Y<4)$

(ii) $\text{P}(X-Y>4)$

A game is played with a biased coin for which the probability of obtaining a tail on any flip is 0.2. In the game, the coin is flipped twice and the random variable $T$ denotes the number of tails obtained.

Draw up the probability distribution table for $T$.

To calculate the score in 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 independent random variables $S$ and $T$ have probability distributions as shown in the following tables.

Find the value of $\text{E}\left(S\right)$.

Find the value of $\text{E}(16S+15T)$.

Given that $\text{Var}\left(T\right)=11.5$, find the value of $\text{Var}(S+2T)$.

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

Find the probability that

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

(ii) the sum of 2 independent observations of $X$ and 3 independent observations of $Y$ is greater than 44.

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

The loading time for a boss fight in a video game is $L$ seconds, and the time taken by Malik to beat the boss is $B$ seconds. $L$ and $B$ have the distributions $\text{N}(14.1,6.1)$ and $\text{N}(101.7,243.5)$ respectively. The total time taken to complete the level is $C$ seconds, where $C=L+B$.

Comment on the assumption that $L$ and $B$ are independent.

Assuming that $L$ and $B$ are independent, state the distribution of $C$, giving the values of any parameters.

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

The independent random variables $X$ and $Y$ have the distributions $\text{Po}(3.5)$ and $\text{B}(16,0.3)$ respectively.

Find the mean and standard deviation of $X-2Y$.

The random variable $X$ has mean 8 and variance 15. The constants $a$ and $b$ are such that

$\text{E}(aX+b)=23$

$\text{Var}(aX+b)=135$

Find the two possible values of $b$.

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

Find the value of:

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

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

(iii) $\text{Var}(100-2X)$

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

The random variable $X$ has mean $\mu$ and standard deviation $\sigma$. The random variable $Y$ has mean 3 and standard deviation 4. It is given that

$\text{E}(2X+5Y)=25$

$\text{Var}(2X+5Y)=724$

Find the value of $\mu$ and the value of $\sigma$.

State the assumption that has been made about the random variables $X$ and $Y$ in your calculation in part (a).

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

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

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

A fair six-sided die has faces numbered 1, 1, 2, 2, 3, 3. The score when the die is thrown is denoted by the random variable $D$.

Show that $\text{E}\left(D\right)=2$ and find $\text{Var}\left(D\right)$.

The die is thrown twice and the arithmetic mean of the two scores is denoted by $\overline{D}$.

Draw up the probability distribution table for $\overline{D}$.

Find $\text{E}\left(\overline{D}\right)$ and $\text{Var}\left(\overline{D}\right)$.

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

Find the probability that Frank's total electricity and gas bill in a randomly chosen month exceeds 150 dollars.

Frank has a part-time job tutoring college students. His monthly income from this job, in dollars, has a normal distribution with mean 504 and standard deviation 41. Frank uses this income to pay his electricity and gas bills, and puts the remaining money into his partner's bank account each month.

(i) Find the probability that Frank puts between 350 dollars and 450 dollars into his partner's bank account in a randomly chosen month.

(ii) State the assumption that was made in part (i) regarding his income and his bills.

Veronica is a taxi driver in London. She charges her customers a fixed fee of 5 dollars plus 1.20 dollars 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 randomly chosen taxi ride costs less than 30 dollars.

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

On a bank holiday, Veronica doubles her prices.

Find the variance of the prices of Veronica's taxi rides on a bank holiday.

Find the probability that a randomly chosen taxi ride on a bank holiday costs more than 60 dollars.

At Dinah's Diner, a triple burger consists of three beef patties, two rashers of bacon and one toasted bread bun. The masses, in grams, of beef patties, rashers of bacon and toasted bread buns are independent and have the distributions $\text{N}(110,6^2)$, $\text{N}(30,5^2)$ and $\text{N}(50,3^2)$ respectively.

Find the probability that a randomly chosen triple burger has a mass of more than 450 g.

Two triple burgers are chosen at random. State, with a reason, whether the probability that their total mass exceeds 900 g is equal to your answer to part (a).

The random variables $X$ and $Y$ are independent. $X$ has mean 2 and variance 3, and $Y$ has mean 5 and variance 6. It is given that

$\text{E}(aX+bY)=-4$

$\text{Var}(aX+bY)=51$

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

The probability that a biased die lands on a 6 on any throw is $p$. In a game, this die is thrown twice. A player's score, $S$, is calculated by multiplying the number of times the die lands on a 6 by 50 and then adding 5.

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

(ii) Find the variance of the scores.

Two friends, Forrest and Gumpy, are planning to run a marathon together. The times, in minutes, taken by Forrest and Gumpy to complete a marathon are modelled by the independent random variables $F$ and $G$ respectively, where $F~\text{N}(253,95)$ and $G~\text{N}(281,52)$.

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 the times taken by Forrest and Gumpy to complete the marathon differ by more than 5 minutes.

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

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

Find the probability that the price of a randomly chosen degu is greater than the price of a randomly chosen rabbit.

Find the probability that the price of a randomly chosen chinchilla is more than five times the price of a randomly chosen rabbit.

Roger and his housemate Lucy decide to buy one rabbit, one chinchilla and one degu. Roger pays the full price of the rabbit, 50% of the price of the chinchilla and 10% of the price of the degu.

Find the probability that the total amount paid by Roger for the three pets is less than 100 dollars.

The random variables $X$ and $Y$ have independent distributions $\text{N}(50,9^2)$ and $\text{N}(400,150)$ respectively.

Find $\text{P}(Y<7X+40)$.

The probability that the sum of a random observation of $X$ and a random observation of $Y$ is greater than $k$ is 0.9995.

Find the value of $k$.

Find the probability that the sum of three independent observations of $X$ is greater than one third of a randomly chosen observation of $Y$.

In a video game, the number of points awarded for completing a level and the number of points awarded for defeating an enemy are modelled by the independent random variables $L~\text{N}(500,220)$ and $E~\text{N}(150,85)$ respectively.

In a bonus stage, the points awarded for completing the level are tripled, and there are 5 enemies. The points awarded for defeating each enemy are independent and are not tripled. The total score in the bonus stage is the sum of the points for completing the level and the points for defeating the 5 enemies.

Scores in the top 10% qualify for the leaderboard.

Estimate the minimum score required to qualify for the leaderboard.

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%.