Expectation and variance

Xian rolls a fair die 10 times. The random variable X represents the number of times the die lands on a six.

Using a suitable distribution for X, find

P(X = 3)

P(X 3) Xian repeats this experiment each day for 60 days and records the number of days when X = 3

Find the probability that there were at least 12 days when X = 3

Find an estimate for the total number of sixes that Xian will roll during these 60 days.

Use a normal approximation to estimate the probability that Xian rolls a total of more than 95 sixes during these 60 days.

A random variable X has probability density function

Use algebraic integration to show that E(X²) =

find, to 3 decimal places, the standard deviation of X

describe, giving a reason, the skewness of the distribution.

Find the cumulative distribution function F(x) for all values of x

Find P(2 X 5)

A random variable X has probability density function

where k and b are constants. Given that E(X í VKRZ WKDW b = 3

The length of time, T, minutes, spent completing a particular task has probability density function

Use algebraic integration to find E(T)

find Var(T)

Find the cumulative distribution function F(t)

Find the 20th percentile of the time taken to complete the task.

Find the probability that the time spent completing the task is more than 1.5 minutes. Given that a person has already spent 1.5 minutes on the task,

find the probability that this person takes more than 3 minutes to complete the task.

The lifetime, X, in tens of hours, of a battery is modelled by the probability density function

Use algebraic integration to find

E(X)

P(X 2.5)

A radio runs using 2 of these batteries, both of which must be working. Two fully-charged batteries are put into the radio.

Find the probability that the radio will be working after 25 hours of use.

Given that the radio is working after 16 hours of use,

find the probability that the radio will be working after being used for another 9 hours.

The random variable R has a continuous uniform distribution over the interval [5, 9]

Specify fully the probability density function of R.

Find P(7 R 10)

The random variable A is the area of a circle radius R cm.

Find E(A)

The weight, X kg, of staples in a bin full of paper has probability density function

Use integration to find

E(X)

Var (X)

P(X 1.5)

Peter raises money by collecting paper and selling it for recycling. A bin full of paper is sold for £50 but if the weight of the staples exceeds 1.5 kg it sells for £25

Find the expected amount of money Peter raises per bin full of paper.

Peter could remove all the staples before the paper is sold but the time taken to remove the staples means that Peter will have 20% fewer bins full of paper to sell.

Decide whether or not Peter should remove all the staples before selling the bins full of paper. Give a reason for your answer.

A continuous random variable X has probability density function f(x) where

where k and n are positive integers.

Find k in terms of n.

Find E(X) in terms of n.

Find E(X²) in terms of n.

Given that n = 2

find Var(3 X).

The random variable X has probability density function f(x) given by

where k is a constant.

Sketch f (x).

Write down the mode of X.

describe, giving a reason, the skewness of the distribution.

Use integration to find the value of k.

Write down the lower quartile of X.

find the exact value of P(X 3).

A piece of string AB has length 9 cm. The string is cut at random at a point P and the random variable X represents the length of the piece of string AP.

Write down the distribution of X.

Find the probability that the length of the piece of string AP is more than 6 cm.

The two pieces of string AP and PB are used to form two sides of a rectangle. The random variable R represents the area of the rectangle.

Show that R = aX² + bX and state the values of the constants a and b.

Find E(R).

Find the probability that R is more than twice the area of a square whose side has the length of the piece of string AP.

The random variable X has probability density function f(x) given by

where k is a constant.

Find the mode of X.

Use algebraic integration to find E(X).

By comparing your answers to parts (b) and (c),

describe the skewness of X, giving a reason for your answer.

The continuous random variable X has a cumulative distribution function

where a and b are constants.

Find the value of a and the value of b.

Use integration to find E(X).

Show that the lower quartile of X lies between 1.425 and 1.435

As part of a selection procedure for a company, applicants have to answer all 20 questions of a multiple choice test. If an applicant chooses answers at random the probability of choosing a correct answer is 0.2 and the number of correct answers is represented by the random variable X.

Suggest a suitable distribution for X.

Each applicant gains 4 points for each correct answer but loses 1 point for each incorrect answer. The random variable S represents the final score, in points, for an applicant who chooses answers to this test at random.

Show that S = 5 X – 20

Find E(S) and Var(S).

An applicant who achieves a score of at least 20 points is invited to take part in the final stage of the selection process.

Find P(S 20)

Cameron is taking the final stage of the selection process which is a multiple choice test consisting of 100 questions. He has been preparing for this test and believes that his chance of answering each question correctly is 0.4

Using a suitable approximation, estimate the probability that Cameron answers more than half of the questions correctly.

A random variable X has the distribution B(12, p).

Given that p = 0.25 find

P(X < 5)

Given that P(X = 0) = 0.05, find the value of p to 3 decimal places.

Given that the variance of X is 1.92, find the possible values of p.

The continuous random variable X has the following probability density function

where a and b are constants.

find a second equation in a and b,

hence find the value of a and the value of b.

Find, to 3 significant figures, the median of X.

Comment on the skewness. Give a reason for your answer.

The continuous random variable X has probability density function given by

Sketch f (x) showing clearly the points where it meets the x-axis.

Write down the value of the mean, of X.

The cumulative distribution function of X is given by

where a is a constant.

Find the value of a.

Show that the lower quartile of X, q₁, lies between 2.29 and 2.31

Hence find the upper quartile of X, giving your answer to 1 decimal place.

Find, to 2 decimal places, the value of k so that

A disease occurs in 3% of a population.

State any assumptions that are required to model the number of people with the disease in a random sample of size n as a binomial distribution.

Using this model, find the probability of exactly 2 people having the disease in a random sample of 10 people.

Find the mean and variance of the number of people with the disease in a random sample of 100 people.

A doctor tests a random sample of 100 patients for the disease. He decides to offer all patients a vaccination to protect them from the disease if more than 5 of the sample have the disease.

Using a suitable approximation, find the probability that the doctor will offer all patients a vaccination.

The continuous random variable X is uniformly distributed over the interval [–1,3]. Find

E(X)

Var(X)

P(X 1.4)

A total of 40 observations of X are made.

Find the probability that at least 10 of these observations are negative.

The queuing time in minutes, X, of a customer at a post office is modelled by the probability density function

Using integration, find

the mean queuing time of a customer,

the probability that a customer will queue for more than 5 minutes.

Three independent customers shop at the post office.

Find the probability that at least 2 of the customers queue for more than 5 minutes.

Bhim and Joe play each other at badminton and for each game, independently of all others, the probability that Bhim loses is 0.2 Find the probability that, in 9 games, Bhim loses

exactly 3 of the games,

fewer than half of the games.

Bhim attends coaching sessions for 2 months. After completing the coaching, the probability that he loses each game, independently of all others, is 0.05 Bhim and Joe agree to play a further 60 games.

Calculate the mean and variance for the number of these 60 games that Bhim loses.

Using a suitable approximation calculate the probability that Bhim loses more than 4 games.

The random variable Y has probability density function f(y) given by

where k and a are positive constants.

Given that E(Y) = 1.75

show that a = 4 and write down the value of k.

For these values of a and k,

sketch the probability density function,

write down the mode of Y.

A manufacturer supplies DVD players to retailers in batches of 20. It has 5% of the players returned because they are faulty.

Write down a suitable model for the distribution of the number of faulty DVD players in a batch.

Find the probability that a batch contains

no faulty DVD players,

more than 4 faulty DVD players.

Find the mean and variance of the number of faulty DVD players in a batch.

A continuous random variable X has cumulative distribution function

Find P(X < 0).

Find the probability density function f(x) of X.

Write down the name of the distribution of X.

Find the mean and the variance of X.

Write down the value of P(X = 1).

The continuous random variable X has probability density function f(x) given by

where k is a constant.

Find the cumulative distribution function F(x).

Find the mean of X.

Show that the median of X lies between x = 2.6 and x = 2.7

The three independent random variables A, B and C each has a continuous uniform distribution over the interval [0, 5].

Find P(A > 3).

Find the probability that A, B and C are all greater than 3.

The random variable Y represents the maximum value of A, B and C. The cumulative distribution function of Y is

Find the probability density function of Y.

Sketch the probability density function of Y.

Write down the mode of Y.

Find E(Y).

Find P(Y > 3).

Figure 1 shows a sketch of the probability density function f(x) of the random variable X. The part of the sketch from x = 0 to x = 4 consists of an isosceles triangle with maximum at (2, 0.5).

Write down E(X).

The probability density function f(x) can be written in the following form.

Find the values of the constants a and b.

Show that , the standard deviation of X, is 0.816 to 3 decimal places.

Find the lower quartile of X.

State, giving a reason, whether P(2 – X is more or less than 0.5

The length of a telephone call made to a company is denoted by the continuous random variable T. It is modelled by the probability density function

Show that the value of k

Find P(T > 6).

Calculate an exact value for E(T) and for Var(T).

Write down the mode of the distribution of T.

It is suggested that the probability density function, f(t), is not a good model for T.

Sketch the graph of a more suitable probability density function for T.

The score when a spinner is spun is given by the discrete random variable X with the following probability distribution, where a and b are probabilities.

Explain why E(X) = 2

Find a linear equation in a and b. Given that Var(X) = 7.1

find a second equation in a and b and simplify your answer.

Solve your two equations to find the value of a and the value of b. The discrete random variable Y = 10 – 3X

(e) Find (i) E(Y)

Var(Y) The spinner is spun once.

Find P(Y X).

The score, X, for a biased spinner is given by the probability distribution

E(X)

Var(X)

A biased coin has one face labelled 2 and the other face labelled 5 The score, Y, when the coin is spun has P(Y = 5) = p and E(Y) = 3

Form a linear equation in p and show that p =

Write down the probability distribution of Y.

Sam plays a game with the spinner and the coin. Each is spun once and Sam calculates his score, S, as follows

if X  0 then S = XY

Find the probability distribution of S.

Find E(S).

Charlotte also plays the game with the spinner and the coin. Each is spun once and Charlotte ignores the score on the coin and just uses X² as her score. Sam and Charlotte each play the game a large number of times.

State, giving a reason, which of Sam and Charlotte should achieve the higher total score.

The discrete random variable X has the following probability distribution, where p and q are constants.

Write down an equation in p and q

Given that E(X) = 0.4

find the value of q

hence find the value of p

Given also that E(X²) = 2.275

find Var(X)

Sarah and Rebecca play a game. A computer selects a single value of X using the probability distribution above. Sarah’s score is given by the random variable S = X and Rebecca’s score is given by the

Find E(R)

Sarah and Rebecca work out their scores and the person with the higher score is the winner. If the scores are the same, the game is a draw.

Find the probability that

Sarah is the winner,

Rebecca is the winner.

In a quiz, a team gains 10 points for every question it answers correctly and loses 5 points for every question it does not answer correctly. The probability of answering a question correctly is 0.6 for each question. One round of the quiz consists of 3 questions. The discrete random variable X represents the total number of points scored in one round. The table shows the incomplete probability distribution of X

Show that the probability of scoring 15 points in a round is 0.432

Find the probability of scoring 0 points in a round.

Find the probability of scoring a total of 30 points in 2 rounds.

Find E(X)

Find Var(X)

In a bonus round of 3 questions, a team gains 20 points for every question it answers correctly and loses 5 points for every question it does not answer correctly.

Find the expected number of points scored in the bonus round.

A biased die with six faces is rolled. The discrete random variable X represents the score on the uppermost face. The probability distribution of X is shown in the table below.

Given that E(X) = 4.2 find the value of a and the value of b.

Show that E(X²) = 20.4

Find Var(5 – 3 X)

A biased die with five faces is rolled. The discrete random variable Y represents the score which is uppermost. The cumulative distribution function of Y is shown in the table

Find the value of k.

Find the probability distribution of Y.

Each die is rolled once. The scores on the two dice are independent.

Find the probability that the sum of the two scores equals 2

The score S when a spinner is spun has the following probability distribution.

Find E(S).

Show that E(S²) = 10.4

Hence find Var(S).

Find

E(5 S – 3),

Var(5 S – 3).

Find P(5 S – 3 > S + 3)

The spinner is spun twice. The score from the first spin is S₁ and the score from the second spin is S₂ The random variables S₁ and S₂ are independent and the random variable X = S₁ × S₂

Show that P({S₁ = 1} X < 5) = 0.16

Find P(X < 5).

The discrete random variable X can take only the values 1, 2 and 3. For these values the cumulative distribution function is defined by

Show that k = 13

Find the probability distribution of X.

find the exact value of Var(4 X – 5).

A fair blue die has faces numbered 1, 1, 3, 3, 5 and 5. The random variable B represents the score when the blue die is rolled.

Write down the probability distribution for B.

State the name of this probability distribution.

Write down the value of E(B).

A second die is red and the random variable R represents the score when the red die is rolled. The probability distribution of R is

Find E(R).

Find Var(R).

Tom invites Avisha to play a game with these dice. Tom spins a fair coin with one side labelled 2 and the other side labelled 5. When Avisha sees the number showing on the coin she then chooses one of the dice and rolls it. If the number showing on the die is than the number showing on the coin, Avisha wins, otherwise Tom wins. Avisha chooses the die which gives her the best chance of winning each time Tom spins the coin.

Find the probability that Avisha wins the game, stating clearly which die she should use in each case.

A discrete random variable X has the probability function

Find E(X)

Find Var(1 – 3 X)

A spinner is designed so that the score S is given by the following probability distribution.

Find the value of p.

Find E(S).

Find Var(S).

Tom and Jess play a game with this spinner. The spinner is spun repeatedly and S counters are awarded on the outcome of each spin. If S is even then Tom receives the counters and if S is odd then Jess receives them. The first player to collect 10 or more counters is the winner.

Find the probability that Jess wins after 2 spins.

Find the probability that Tom wins after exactly 3 spins.

Find the probability that Jess wins after exactly 3 spins.

The discrete random variable X has the probability distribution

Show that k = 0.1

Find

E(X)

Var(2 – 5 X)

Two independent observations X₁ and X₂ are made of X.

Show that P(X₁ + X₂ = 4) = 0.1

Complete the probability distribution table for X₁ + X₂

The discrete random variable X has probability distribution given by

x	1	0	1	2	3
P(X x)	1 5	a	1 10	a	1 5

where a is a constant.

Find the value of a.

Write down E(X).

Find Var(X).

Find Var(Y).

The discrete random variable X has probability function

Find P(X = 2) and complete the table below.

Given that E(X) = 1.6

Find the value of a and the value of b.

Find

P(0.5 < X < 3),

E(3 X – 2).

Show that the Var(X) = 1.64

Calculate Var(3 X – 2).

When Rohit plays a game, the number of points he receives is given by the discrete random variable X with the following probability distribution.

Find E(X).

Find F(1.5).

Show that Var(X) = 1

Find Var(5 – 3 X).

Rohit can win a prize if the total number of points he has scored after 5 games is at least 10. After 3 games he has a total of 6 points. You may assume that games are independent.

Find the probability that Rohit wins the prize.

The random variable X has probability distribution given in the table below.

Given that E(X) = 0.55, find

the value of p and the value of q,

Var(X),

E(2 X – 4).

Tetrahedral dice have four faces. Two fair tetrahedral dice, one red and one blue, have faces numbered 0, 1, 2, and 3 respectively. The dice are rolled and the numbers face down on the two dice are recorded. The random variable R is the score on the red die and the random variable B is the score on the blue die.

Find P(R =3 and B =0).

The random variable T is R multiplied by B.

Complete the diagram below to represent the sample space that shows all the possible values of T.

The table below represents the probability distribution of the random variable T.

Find the values of a, b, c and d.

Find the values of

E(T),

Var(T).