5.02b Expectation and variance: discrete random variables

1. A multiple-choice test consists of 25 questions, each having 5 responses, only one of which is correct.

Each correct answer gains 4 marks but each incorrect answer loses 1 mark.
Sam answers all 25 questions by choosing at random one response for each question.
Let \(X\) be the number of correct answers that Sam achieves.

1. State the distribution of \(X\) Let \(M\) be the number of marks that Sam achieves.
2. 1. State the distribution of \(M\) in terms of \(X\)
   2. Hence, show clearly that the number of marks that Sam is expected to achieve is zero. In order to pass the test at least 30 marks are required.
3. Find the probability that Sam will pass the test. Past records show that when the test is done properly, the probability that a student answers the first question correctly is 0.5 A random sample of 50 students that did the test properly was taken.
   Given that the probability that more than \(n\) but at most 30 students answered the first question correctly was 0.9328 to 4 decimal places,
4. find the value of \(n\)

6 James plays an arcade game. Each time he plays, he puts a \(\pounds 1\) coin in the slot to start the game. The possible outcomes of each game are as follows: James loses the game with a probability of 0.7 and the machine pays out nothing, James draws the game with a probability of 0.25 and the machine pays out a \(\pounds 1\) coin, James wins the game with a probability of 0.05 and the machine pays out ten \(\pounds 1\) coins. The outcomes can be modelled by a random variable \(X\) representing the number of \(\pounds 1\) coins gained at the end of a game.

1. Construct a probability distribution table for \(X\).
2. Show that \(\mathrm { E } ( X ) = - 0.25\) and find \(\operatorname { Var } ( X )\). James starts off with \(10 \pounds 1\) coins and decides to play exactly 10 games.
3. Find the expected number of \(\pounds 1\) coins that James will have at the end of his 10 games.
4. Find the probability that after his 10 games James will have at least \(10 \pounds 1\) coins left.

3 The probability generating function of the random variable \(X\) is \(\frac { 1 } { 81 } \left( t + \frac { 2 } { t } \right) ^ { 4 }\).

1. Use the probability generating function to find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
2. The random variable \(Y\) is defined by \(Y = \frac { 1 } { 2 } ( X + 4 )\). By finding the probability distribution of \(X\), or otherwise, show that \(Y \sim \mathrm {~B} ( n , p )\), stating the values of \(n\) and \(p\).

3

1. Show that the probability generating function of a random variable with the distribution \(\mathrm { B } ( n , p )\) is \(( 1 - p + p t ) ^ { n }\).
2. \(R\) and \(S\) are independent random variables with the distributions \(\mathrm { B } \left( 8 , \frac { 1 } { 4 } \right)\) and \(\mathrm { B } \left( 8 , \frac { 3 } { 4 } \right)\) respectively. Show that the probability generating function of \(R + S\) can be expressed as $$\left( \frac { 3 } { 16 } + \frac { 1 } { 16 } t ( 10 + 3 t ) \right) ^ { 8 }$$ and use this result to find \(\mathrm { P } ( R + S = 1 )\).

3 The probability distribution of the discrete random variable \(X\) is defined as follows. $$\mathrm { P } ( X = x ) = k ( 2 + x ) ( 5 - x ) \quad \text { for } x = 0,1,2,3,4$$

1. Show that \(k = \frac { 1 } { 50 }\).
2. Find the variance of \(X\).
3. Find \(\mathrm { P } ( X = 4 \mid X > 0 )\).

3 John plays a game with two unbiased coins. John tosses the coins. If he gets two heads he wins \(\pounds 1\). If he gets two tails he wins 20 p. If he gets one head and one tail he wins nothing. Let \(X\) be the random variable for the amount of money, in pence, John wins per game.

1. Construct a probability distribution table for \(X\).
2. Calculate \(\mathrm { E } ( X )\).
3. John pays \(s\) pence to play the game. State the values of \(s\) for which John should expect to make a loss.

1 The discrete random variable X has probability generating function \(\mathrm { G } _ { X } ( t )\) given by $$G _ { X } ( t ) = a t \left( t + \frac { 1 } { t } \right) ^ { 3 }$$ where \(a\) is a constant.

1. Find, in either order, the value of \(a\) and the set of values that \(X\) can take.
2. Find the value of \(\mathrm { E } ( X )\).

12 A faulty random number generator generates odd digits according to the probability distribution for the random variable \(X\) given in the following table.

1. Find \(p\).
2. Find \(\mathrm { E } ( X )\) and \(\mathrm { E } \left( X ^ { 2 } \right)\).
3. Deduce the value of \(\operatorname { Var } ( X )\). A second random number generator generates odd digits each with equal probability. Both random generators are operated once.
4. Find the probability that both generate a prime number.
5. Given that the first generates 1, 3 or 5, find the probability that both generate a power of 3 . 1315 pupils, including two sisters, are placed in random order in a line.
6. What is the probability that the sisters are not next to each other?
7. How many arrangements are there with 9 pupils between the sisters? A team of 5 is chosen from the 15 pupils.
8. How many ways are there of choosing the team if no more than one of the sisters can be in the team? Having chosen the first team, a second team of 5 pupils is chosen from the remaining 10 pupils.
9. How many ways are there of choosing the teams if each sister is in one or other of the teams?

12 A game is played in which the number of points scored, \(X\), has the probability distribution given in the following table.

1. Write down the probability generating function of \(X\).
2. Use this generating function to find the mean and variance of \(X\).
3. The game is played 4 times (independently) and the total number of points scored is denoted by \(Y\). Show that the probability generating function of \(Y\) can be written in the form $$\frac { \left( a + t ^ { 2 } \right) ^ { 8 } } { b t ^ { 4 } }$$ where \(a\) and \(b\) are constants.
4. Hence find \(\mathrm { P } ( Y < 0 )\).

Sharik attempts a multiple choice revision question on-line. There are 3 suggested answers, one of which is correct. When Sharik chooses an answer the computer indicates whether the answer is right or wrong. Sharik first chooses one of the three suggested answers at random. If this answer is wrong he has a second try, choosing an answer at random from the remaining 2. If this answer is also wrong Sharik then chooses the remaining answer, which must be correct.

1. Draw a fully labelled tree diagram to illustrate the various choices that Sharik can make until the computer indicates that he has answered the question correctly. [4]
2. The random variable \(X\) is the number of attempts that Sharik makes up to and including the one that the computer indicates is correct. Draw up the probability distribution table for \(X\) and find E\((X)\). [4]

\(X\) is a discrete random variable which takes the values 0, 2, 4, ... . The probability generating function of \(X\) is given by $$G_X(t) = \frac{1}{3 - 2t^2}.$$

1. Find E\((X)\) and Var\((X)\). [5]

A discrete random variable \(Y\) has probability function $$\mathrm{P}(Y = y) = \begin{cases} k(3 - y) & y = 1, 2 \\ k(y^2 - 8) & y = 3, 4, 5 \\ k & y = 6 \\ 0 & \text{otherwise} \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac{1}{30}\) [2]

Find the exact value of

1. P\((1 < Y \leqslant 4)\) [2]
2. E\((Y)\) [2]

The random variable \(X = 15 - 2Y\)

1. Calculate P\((Y \geqslant X)\) [3]
2. Calculate Var\((X)\) [4]

A discrete random variable \(X\) has the probability function shown in the table below.

1. Given that E(\(X\)) = \(\frac{2}{3}\), find \(a\). [3]
2. Find the exact value of Var (\(X\)). [3]
3. Find the exact value of P(\(X \leq 15\)). [1]

The probability function of a discrete random variable \(X\) is given by $$p(x) = kx^2 \quad x = 1, 2, 3$$ where \(k\) is a positive constant.

1. Show that \(k = \frac{1}{14}\) [2]

Find

1. P\((X \geq 2)\) [2]
2. E\((X)\) [2]
3. Var\((1-X)\) [4]

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

1. Find the value of \(p\). [2]
2. Find \(\text{E}(S)\). [2]
3. Show that \(\text{E}(S^2) = 9.45\) [2]
4. Find \(\text{Var}(S)\). [2]

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.

1. Find the probability that Jess wins after 2 spins. [2]
2. Find the probability that Tom wins after exactly 3 spins. [4]
3. Find the probability that Jess wins after exactly 3 spins. [3]

The discrete random variable \(X\) has the following probability distribution.

1. Given that \(E(X) = -0.2\), find the value of \(\alpha\) and the value of \(\beta\). [6]
2. Write down \(F(0.8)\). [1]
3. Evaluate \(\text{Var}(X)\). [4]

Find the value of

1. \(E(3X - 2)\), [2]
2. \(\text{Var}(2X + 6)\). [2]

The discrete random variable \(X\) has probability function $$P(X = x) = \begin{cases} kx, & x = 1, 2, 3, 4, 5, \\ 0, & \text{otherwise.} \end{cases}$$

1. Show that \(k = \frac{1}{15}\). [3]

Find the value of

1. E\((2X + 3)\), [5]
2. Var\((2X - 4)\). [6]

In a town, 30\% of residents listen to the local radio station. Four residents are chosen at random.

1. State the distribution of the random variable \(X\), the number of these four residents that listen to local radio. [2]
2. On graph paper, draw the probability distribution of \(X\). [3]
3. Write down the most likely number of these four residents that listen to the local radio station. [1]
4. Find E(\(X\)) and Var (\(X\)). [3]

1. State two conditions under which a random variable can be modelled by a binomial distribution. [2]

In the production of a certain electronic component it is found that 10% are defective. The component is produced in batches of 20.

1. Write down a suitable model for the distribution of defective components in a batch. [1]

Find the probability that a batch contains

1. no defective components, [2]
2. more than 6 defective components. [2]
3. Find the mean and the variance of the defective components in a batch. [2]

A supplier buys 100 components. The supplier will receive a refund if there are more than 15 defective components.

1. Using a suitable approximation, find the probability that the supplier will receive a refund. [4]

The discrete random variable \(X\) can take any value in the set \(\{1, 2, 3, 4, 5, 6, 7, 8\}\). Arthur, Beatrice and Chris each carry out trials to investigate the distribution of \(X\). Arthur finds that P\((X = 1) = 0.125\) and that E\((X) = 4.5\). Beatrice finds that P\((X = 2) =\) P\((X = 3) =\) P\((X = 4) = p\). Chris finds that the values of \(X\) greater than 4 are all equally likely, with each having probability \(q\).

1. Calculate the values of \(p\) and \(q\). [7 marks]
2. Give the name for the distribution of \(X\). [1 mark]
3. Calculate the standard deviation of \(X\). [3 marks]

The discrete random variable \(X\) has the probability function given by the following table:

1. Show that \(p = 0.07\) [2 marks]
2. Find the value of \(E(X + 2)\). [4 marks]
3. Find the value of \(\text{Var}(3X - 1)\). [5 marks]

Thirty cards, marked with the even numbers from 2 to 60 inclusive, are shuffled and one card is withdrawn at random and then replaced. The random variable \(X\) takes the value of the number on the card each time the experiment is repeated.

1. What must be assumed about the cards if the distribution of \(X\) is modelled by a discrete uniform distribution? [1 mark]
2. Making this modelling assumption, find the expectation and the variance of \(X\). [5 marks]

The discrete random variable \(X\) has probability function P\((X = x) = k(x + 4)\). Given that \(X\) can take any of the values \(-3, -2, -1, 0, 1, 2, 3, 4\),

1. find the value of the constant \(k\). [3 marks]
2. Find P\((X < 0)\). [2 marks]
3. Show that the cumulative distribution F\((x)\) is given by $$\text{F}(x) = \lambda(x + 4)(x + 5)$$ where \(\lambda\) is a constant to be found. [6 marks]

The discrete random variable \(X\) has probability function $$P(X = x) = \begin{cases} cx^2 & x = -3, -2, -1, 1, 2, 3 \\ 0 & \text{otherwise.} \end{cases}$$

1. Show that \(c = \frac{1}{28}\). [3 marks]
2. Calculate
   1. \(E(X)\),
   2. \(E(X^2)\).
   [3 marks]
3. Calculate
   1. \(\text{Var}(X)\),
   2. \(\text{Var}(10 - 2X)\).
   [3 marks]