Discrete random variables

The discrete random variable R takes even integer values from 2 to 2n inclusive. The probability distribution of R is given by

where k is a constant.

Show that k = n(n + 1) When n = 20

find the exact value of P (16 ≤ R < 26) When n = 20, a random value g of R is taken and the quadratic equation in x

is formed.

Find the exact probability that the equation has no real roots.

TOTAL FOR STATISTICS IS 50 MARKS

Tisam is playing a game. She uses a ball, a cup and a spinner. The random variable X represents the number the spinner lands on when it is spun. The probability distribution of X is given in the following table

where a, b, c and d are probabilities. To play the game

• the spinner is spun to obtain a value of x

• Tisam then stands x cm from the cup and tries to throw the ball into the cup

The event S represents the event that Tisam successfully throws the ball into the cup. To model this game Tisam assumes that

• P(S {X = x}) should be the same whatever value of x is obtained from the spinner

Using Tisam’s model,

find the probability distribution of X Nav tries, a large number of times, to throw the ball into the cup from a distance of 100 cm. He successfully gets the ball in the cup 30% of the time.

State, giving a reason, why Tisam’s model of this game is not suitable to describe Nav playing the game for all values of X

The discrete random variable X has the following probability distribution

• a, b and c are distinct integers (a < b < c)

• all the probabilities are greater than zero

Find

the value of a

the value of b

the value of c Show your working clearly. The independent random variables X₁ and X₂ each have the same distribution as X

Find P(X₁ = X₂)

TOTAL FOR STATISTICS IS 50 MARKS

The discrete random variable D has the following probability distribution

where k is a constant.

Show that the value of k is

The random variables D₁ and D₂ are independent and each have the same distribution as D.

Find P (D₁ + D₂ = 80) Give your answer to 3 significant figures. A single observation of D is made. The value obtained, d, is the common difference of an arithmetic sequence. The first 4 terms of this arithmetic sequence are the angles, measured in degrees, of quadrilateral Q

Find the exact probability that the smallest angle of Q is more than 50°

The discrete random variable X has the following probability distribution

where a, b and c are probabilities. The cumulative distribution function of X is F(x) and F(3) = 0.2 and F(6) = 0.8

Find the value of a, the value of b and the value of c.

Write down the value of F(7).

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 discrete random variable X has probability distribution

The cumulative distribution function of X is given by

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

Write down the value of P(X² = 1).

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.

The discrete random variable X takes the values 1, 2 and 3 and has cumulative distribution function F(x) given by

Find the probability distribution of X.

Write down the value of F(1.8).

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 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)

The discrete random variable Y has probability distribution

where a, b and c are constants. The cumulative distribution function F(y) of Y is given in the following table

where d is a constant.

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

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 discrete random variable X can take only the values 2, 3 or 4. For these values the cumulative distribution function is defined by

where k is a positive integer.

Find k.

Find the probability distribution of X.

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).