Random Variables (DP IB Maths: AI HL)

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

find the two possible values of .

The random variable has mean of  and standard deviation of .  The random variable  has mean of 3 and standard deviation of 4.  Given that

Find the value of  and the value of .

State the assumption that has been made about the random variables  and .

Frank has a variable tariff for his electricity and gas bills. His monthly electricity bill is $E  and his monthly gas bill is $G .   and  are independent random variables with distributions  and  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.

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 has the distribution .

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

Hence find the probability that the mean of 50 independent observations of 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 .  The mass, in grams, of a rasher of bacon follows the distribution .  The mass, in grams, of a toasted bread bun follows the distribution .

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).

Ella buys some fruit from the grocery store. The average mass of an apple at the store is 220 grams with standard deviation 15 grams. The average mass of an orange at the store is 120 grams with standard deviation 8 grams. Ella buys 5 random apples and 8 random oranges and packs them in her grocery bag which weighs 135 grams when empty.

Find the expectation and standard deviation of the total mass of the grocery bag and the 13 pieces of fruit. State any assumptions that are needed and where they are needed.

A football coach ends each training session with a penalty shoot-out competition. Each player takes 15 penalty shots and scores 6 points for each goal. The coach does not want anybody to get zero points so gives all players 10 points just for participating. Raquel takes part in the challenge each session and it is known that on average 65% of her shots go in the goal.

Find the mean and standard deviation for the number of points Raquel achieves in the competition. State any assumptions that are needed.

Find the probability that Raquel scores more than 80 points in the competition.

Harietta has a summer job during her break from college. The random variable  represents the amount of money ($) Harietta earns each day. Each day she gets a guaranteed $50 plus an extra $10 for every extra hour she works. The number of extra hours she works each day can be modelled by the random variable . The probability distribution of  is shown below.

	0	1	2	3	4
	0.35		0.07		0.18

Given that the expected amount of money that Harietta earns in a day is $65.20, find the values of and .

Given that find .

Two friends, Forrest and Gumpy, are planning to run a marathon together. The distributions  and  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.

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  and  are independent.

Find .

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

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

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  and  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.

Kate and Clint are working as a pair in an archery competition. They are both shooting arrows at a target. Kate shoots 20 arrows and Clint shoots 10. The number of times they each hit the target are added together to form the pair’s final score, denoted by the random variable . On average, Kate has an 95% chance of hitting the target and Clint has a 50% chance of hitting the target.

Find . State any assumptions that are needed.

Find . State an additional assumption that is needed.

Kate claims that the pair’s final score,, follows a binomial distribution .

By using the formulae for the mean and variance of a binomial distribution, show that Kate’s claim is incorrect.

In the competition, pairs win a prize if their final score is at least 28.

Find the probability that Kate and Clint win a prize.

Viktor works for the emergency services and has found from previous data that the amount of call-outs per day can be modelled using a Poisson distribution with mean 14.9. The number of call-outs are independent of the day of the week. Viktor decides to monitor the number of call-outs each day over a seven-day period.

Find the mean and the standard deviation of the total number of call-outs during a seven-day period.

Find the probability that the mean number of daily call-outs using Viktor’s seven-day period is more than 16.

After each call-out Viktor is required to complete three forms. 

Find the mean and standard deviation of the number of forms that Viktor is required to complete in a day due to call-outs.

Reuben works at a candy store and sells three types of sweets: chocolate, marshmallow and honeycomb. Reuben uses a scoop to measure a portion for each type of sweet and the price depends on the weight of each individual portion. The table below shows the mean and standard deviation of the masses of the portions for each type of sweet as well as the cost per unit weight. 

Reuben offers a product called Sugar Supreme which contains 10 portions of sweets in total. Two portions are chocolate,  portions are marshmallow and  portions are honeycomb. The mean cost of a Sugar Supreme is £3.85.

Find the values of and .

Find the standard deviation of the costs of the Sugar Supreme product.

Danny and Mark use a biased four-sided dice to play a game. The number that the dice lands on, , follows the probability distribution described in the table below.

	0	2	6	8

Danny calculates his score by multiplying the number on the dice by 15 and then adding 11. Mark calculates his score by adding 5 to the number on the dice and then multiplying by 8. They each roll the dice once and calculate their scores.

Given that Mark’s expected score is 8 more than Danny’s expected score, find the values of and .

After they have rolled the dice once, each player is awarded a number of points which is calculated by subtracting their opponent’s score from their own score. A player’s number of points will be negative if their opponent’s score is higher than their own.

Given that the standard deviation for the number of points a player is awarded is 51, calculate the standard deviation of .