5.02b Expectation and variance: discrete random variables

A spinner has edges numbered 1, 2, 3, 4 and 5. When the spinner is spun, the number of the edge on which it lands is the score. The probability distribution of the score, \(N\), is given in the table. It is known that E\((N) = 2.55\).

1. Find Var\((N)\). [7]
2. Find E\((3N + 2)\). [1]
3. Find Var\((3N + 2)\). [1]

The discrete random variable \(D\) has the following probability distribution where \(k\) is a constant.

1. Show that the value of \(k\) is \(\frac{600}{137}\) [2]
2. The random variables \(D_1\) and \(D_2\) are independent and each have the same distribution as \(D\). Find \(P(D_1 + D_2 = 80)\) Give your answer to 3 significant figures. [3]
3. 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 \(50°\) [5]

The discrete random variable \(X\) takes values 1, 2, 3, 4 and 5, and its probability distribution is defined as follows. $$\mathrm{P}(X = x) = \begin{cases} a & x = 1, \\ \frac{1}{2}\mathrm{P}(X = x - 1) & x = 2, 3, 4, 5, \\ 0 & \text{otherwise,} \end{cases}$$ where \(a\) is a constant.

1. Show that \(a = \frac{16}{31}\). [2]

The discrete probability distribution for \(X\) is given in the table.

1. Find the probability that \(X\) is odd. [1]

Two independent values of \(X\) are chosen, and their sum \(S\) is found.

1. Find the probability that \(S\) is odd. [2]
2. Find the probability that \(S\) is greater than 8, given that \(S\) is odd. [3]

Sheila sometimes needs several attempts to start her car in the morning. She models the number of attempts she needs by the discrete random variable \(Y\) defined as follows. $$\mathrm{P}(Y = y + 1) = \frac{1}{2}\mathrm{P}(Y = y) \quad \text{for all positive integers } y.$$

1. Find P\((Y = 1)\). [2]
2. Give a reason why one of the variables, \(X\) or \(Y\), might be more appropriate as a model for the number of attempts that Sheila needs to start her car. [1]

At a factory that makes crockery the quality control department has found that 10\% of plates have minor faults. These are classed as 'seconds'. Plates are stored in batches of 12. The number of seconds in a batch is denoted by \(X\).

1. State an appropriate distribution with which to model \(X\). Give the value(s) of any parameter(s) and state any assumptions required for the model to be valid. [4]

Assume now that your model is valid.

1. Find
   1. P\((X = 3)\), [2]
2. A random sample of 4 batches is selected. Find the probability that the number of these batches that contain at least 1 second is fewer than 3. [4]

The probability distribution for the discrete random variable \(W\) is given in the table.

\(w\)	1	2	3	4
\(P(W = w)\)	0.25	0.36	\(x\)	\(x^2\)

1. Show that \(\text{Var}(W) = 0.8571\). [7]
2. Find \(\text{Var}(3W + 6)\). [1]

A random variable \(X\) has probability distribution given by \(P(X = x) = \frac{1}{860}(1 + x)\) for \(x = 1, 2, 3, \ldots, 40\).

1. Find \(P(X > 39)\). [2]
2. Given that \(x\) is even, determine \(P(X < 10)\). [6]

A game is played as follows. A fair six-sided dice is thrown once. If the score obtained is even, the amount of money, in £, that the contestant wins is half the score on the dice, otherwise it is twice the score on the dice.

1. Find the probability distribution of the amount of money won by the contestant. [3]
2. The contestant pays £5 for every time the dice is thrown. Find the standard deviation of the loss made by the contestant in 120 throws of the dice. [5]

The probability distribution of a discrete random variable \(W\) is given in the table. Given that \(\mathrm{E}(W) = 1.61\), find the value of \(\text{Var}(3W + 2)\). [7]

\begin{enumerate}[label=(\alph*)] \item In a game show contestants are asked up to five questions in succession to qualify for the next round. An incorrect answer eliminates a contestant from the game show. Let \(X\) denote the number of questions correctly answered by a contestant. The probability distribution of \(X\) is given below.

\(x\)	0	1	2	3	4	5
\(\mathrm{P}(X = x)\)	0.30	0.25	0.20	0.16	0.06	0.03

1. Find the expected number of correctly answered questions and the variance of the distribution. [3]
2. Find the probability that a randomly selected contestant will correctly answer 3 or more questions. [1]
3. Each show had two contestants. Find the probability that both the contestants will correctly answer at least one question. [2]

\item In a promotion, a newspaper included a token in every copy of the newspaper. A proportion, 0.002, are winning tokens and occur randomly. A reader keeps buying copies of the newspaper until he buys one with a winning token and then stops. Let \(Y\) denote the number of copies bought.

1. Explain briefly why this situation may be modelled by a geometric distribution and write down a formula for \(\mathrm{P}(Y = y)\). [2]
2. Find the probability that the reader gets a winning token with the twentieth copy bought. [2]
3. Find the probability that the reader will not have to buy more than three copies in order to get a winning token. [2] \end{enumerate]

The random variable \(X\) is defined as the difference (always positive or zero) between the scores when 2 ordinary dice are rolled.

1. Copy and complete the probability distribution table for \(X\). [2]

   \(x\)	0	1	2	3	4	5
   P(\(X = x\))

2. Find the expectation and variance of \(X\). [5]

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

\(x\)	1	2	\(n\)	7
\(\mathrm{P}(X = x)\)	0.4	0.3	\(p\)	0.1

1. Write down the value of \(p\). [1]
2. Given that \(\mathrm{E}(X) = 2.5\), find \(n\). [2]
3. Find \(\mathrm{Var}(X)\). [3]

James plays an arcade game. Each time he plays, he puts a £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 £1 coin, James wins the game with a probability of 0.05 and the machine pays out ten £1 coins. The outcomes can be modelled by a random variable \(X\) representing the number of £1 coins gained at the end of a game.

1. Construct a probability distribution table for \(X\). [2]
2. Show that E(\(X\)) = \(-0.25\) and find Var(\(X\)). [4]

James starts off with 10 £1 coins and decides to play exactly 10 games.

1. Find the expected number of £1 coins that James will have at the end of his 10 games. [2]
2. Find the probability that after his 10 games James will have at least 10 £1 coins left. [3]

James plays an arcade game. Each time he plays, he puts a £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 £1 coin, James wins the game with a probability of 0.05 and the machine pays out ten £1 coins. The outcomes can be modelled by a random variable \(X\) representing the number of £1 coins gained at the end of a game.

1. Construct a probability distribution table for \(X\). [2]
2. Show that E(\(X\)) = -0.25 and find Var(\(X\)). [4]

James starts off with 10 £1 coins and decides to play exactly 10 games.

1. Find the expected number of £1 coins that James will have at the end of his 10 games. [2]
2. Find the probability that after his 10 games James will have at least 10 £1 coins left. [3]

A biased tetrahedral die has faces numbered \(1\) to \(4\). The random variable \(X\) is the number on the face of the die which is in contact with the table after the die has been thrown. It is known, for this die, that \(\text{P}(X = x) = kx\) where \(k\) is a constant.

1. Determine the value of \(k\) and state the moment generating function of \(X\). [3]
2. Hence find \(\text{E}(X)\) and \(\text{Var}(X)\). [7]