5.02b Expectation and variance: discrete random variables

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

\(x\)	0	1	2	3	4	5
\(\text{P}(X = x)\)	0.11	0.17	0.2	0.13	\(p\)	\(p^2\)

1. Find the value of \(p\). [4 marks]
2. Find
   1. \(\text{P}(0 < X \leq 2)\),
   2. \(\text{P}(X \geq 3)\).
   [3 marks]
3. Find the mean and the variance of \(X\). [3 marks]
4. Construct a table to represent the cumulative distribution function \(\text{F}(x)\). [2 marks]

The random variable \(X\), which can take any value from \(\{1, 2, \ldots, n\}\), is modelled by the discrete uniform distribution with mean 10.

1. Show that \(n = 19\) and find the variance of \(X\). [4 marks]
2. Find \(\text{P}(3 < X \leq 6)\). [2 marks]

The random variable \(Y\) is defined by \(Y = 3(X - 10)\).

1. State the mean and the variance of \(Y\). [3 marks]

The model for the distribution of \(X\) is found to be unsatisfactory, and in a refined model the probability distribution of \(X\) is taken to be $$\text{f}(x) = \begin{cases} k(x + 1) & x = 1, 2, \ldots, 19, \\ 0 & \text{otherwise}. \end{cases}$$

1. Show that \(k = \frac{1}{209}\). [3 marks]
2. Find \(\text{P}(3 < X \leq 6)\) using this model. [3 marks]

A pack of 52 cards contains 4 cards bearing each of the integers from 1 to 13. A card is selected at random. The random variable \(X\) represents the number on the card.

1. Find \(P(X \leq 5)\). [1 mark]
2. Name the distribution of \(X\) and find the expectation and variance of \(X\). [4 marks]

A hand of 12 cards consists of three 2s, four 3s, two 4s, two 5s and one 6. The random variable \(Y\) represents the number on a card chosen at random from this hand.

1. Draw up a table to show the probability distribution of \(Y\). [3 marks]
2. Calculate \(\text{Var}(3Y - 2)\). [6 marks]

The discrete random variable \(X\) takes only the values \(4, 5, 6, 7, 8\) and \(9\). The probabilities of these values are given in the table:

\(x\)	\(4\)	\(5\)	\(6\)	\(7\)	\(8\)	\(9\)
P\((X = x)\)	\(p\)	\(0.1\)	\(q\)	\(q\)	\(0.3\)	\(0.2\)

It is known that E\((X) = 6.7\). Find

1. the values of \(p\) and \(q\), [7 marks]
2. the value of \(a\) for which E\((2X + a) = 0\), [3 marks]
3. Var\((X)\). [3 marks]

A regular tetrahedron has its faces numbered 1, 2, 3 and 4. It is weighted so that when it is thrown, the probability of each face being in contact with the table is inversely proportional to the number on that face. This number is represented by the random variable \(X\).

1. Show that \(P(X = 1) = \frac{12}{25}\) and find the probabilities of the other values of \(X\). [5 marks]
2. Calculate the mean and the variance of \(X\). [5 marks]

A certain four-sided die is biased. The score, \(X\), on each throw is a random variable with probability distribution as shown in the table. Throws of the die are independent.

\(x\)	0	1	2	3
P\((X = x)\)	\(\frac{1}{2}\)	\(\frac{1}{4}\)	\(\frac{1}{8}\)	\(\frac{1}{8}\)

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

The die is thrown 10 times.

1. Find the probability that there are not more than 4 throws on which the score is 1. [2]
2. Find the probability that there are exactly 4 throws on which the score is 2. [3]

When a four-sided spinner is spun, the number on which it lands is denoted by \(X\), where \(X\) is a random variable taking values 2, 4, 6 and 8. The spinner is biased so that P(\(X = x\)) = \(kx\), where \(k\) is a constant.

1. Show that P(\(X = 6\)) = \(\frac{3}{10}\). [2]
2. Find E(\(X\)) and Var(\(X\)). [5]

20% of packets of a certain kind of cereal contain a free gift. Jane buys one packet a week for 8 weeks. The number of free gifts that Jane receives is denoted by \(X\). Assuming that Jane's 8 packets can be regarded as a random sample, find

1. P(\(X = 3\)), [3]
2. P(\(X \geqslant 3\)), [2]
3. E(\(X\)). [2]

A game at a charity event uses a bag containing 19 white counters and 1 red counter. To play the game once a player takes counters at random from the bag, one at a time, without replacement. If the red counter is taken, the player wins a prize and the game ends. If not, the game ends when 3 white counters have been taken. Niko plays the game once.

1. 1. Copy and complete the tree diagram showing the probabilities for Niko. [4] \includegraphics{figure_2}
   2. Find the probability that Niko will win a prize. [3]
2. The number of counters that Niko takes is denoted by \(X\).
   1. Find P(\(X = 3\)). [2]
   2. Find E(\(X\)). [4]

Repeated independent trials of a certain experiment are carried out. On each trial the probability of success is 0.12.

1. Find the smallest value of \(n\) such that the probability of at least one success in \(n\) trials is more than 0.95. [3]
2. Find the probability that the 3rd success occurs on the 7th trial. [5]

Each of four cards has a number printed on it as shown. Two of the cards are chosen at random, without replacement. The random variable \(X\) denotes the sum of the numbers on these two cards.

1. Show that P\((X = 6) = \frac{1}{6}\) and P\((X = 4) = \frac{1}{3}\). [3]
2. Write down all the possible values of \(X\) and find the probability distribution of \(X\). [4]
3. Find E\((X)\) and Var\((X)\). [5]

There are 10 numbers in a list. The first 9 numbers have mean 6 and variance 2. The 10th number is 3. Find the mean and variance of all 10 numbers. [6]

The probability distribution of a random variable \(X\) is shown.

\(x\)	1	3	5	7
P\((X = x)\)	0.4	0.3	0.2	0.1

1. Find E\((X)\) and Var\((X)\). [5]
2. Three independent values of \(X\), denoted by \(X_1\), \(X_2\) and \(X_3\), are chosen. Given that \(X_1 + X_2 + X_3 = 19\), write down all the possible sets of values for \(X_1\), \(X_2\) and \(X_3\) and hence find P\((X_1 = 7)\). [2]
3. 11 independent values of \(X\) are chosen. Use an appropriate formula to find the probability that exactly 4 of these values are 5s. [3]

In her purse, Katharine has two £5 notes, two £10 notes and one £20 note. She decides to select two of these notes at random to donate to a charity. The total value of these two notes is denoted by the random variable \(£X\).

1. 1. Show that P(X = 10) = 0.1. [1]
   2. Show that P(X = 30) = 0.2. [2]
   The table shows the probability distribution of X.

   \(r\)	10	15	20	25	30
   P(X = r)	0.1	0.4	0.1	0.2	0.2

2. Find E(X) and Var(X). [5]

The probability distribution of the random variable \(X\) is given by the formula $$\text{P}(X = r) = kr(r + 1) \quad \text{for } r = 1, 2, 3, 4, 5.$$

1. Show that \(k = \frac{1}{70}\). [2]
2. Find E\((X)\) and Var\((X)\). [5]

The probability distribution of the random variable \(X\) is given by the formula $$\mathrm{P}(X = r) = k + 0.01r^2 \text{ for } r = 1, 2, 3, 4, 5.$$

1. Show that \(k = 0.09\). Using this value of \(k\), display the probability distribution of \(X\) in a table. [3]
2. Find \(\mathrm{E}(X)\) and \(\mathrm{Var}(X)\). [5]

A six-sided die is biased such that there is an equal chance of scoring each of the numbers from 1 to 5 but a score of 6 is three times more likely than each of the other numbers.

1. Write down the probability distribution for the random variable, \(X\), the score on a single throw of the die. [4]
2. Show that E\((X) = \frac{33}{8}\). [3]
3. Find E\((4X - 1)\). [2]
4. Find Var\((X)\). [4]

The discrete random variable \(X\) has the probability function shown below. $$P(X = x) = \begin{cases} kx, & x = 2, 3, 4, 5, 6, \\ 0, & \text{otherwise}. \end{cases}$$

1. Find the value of \(k\). [2 marks]
2. Show that E\((X) = \frac{9}{2}\). [3 marks]

Find

1. P\([X > \text{E}(X)]\), [2 marks]
2. E\((2X - 5)\), [2 marks]
3. Var\((X)\). [4 marks]

A netball team are in a league with three other teams from which one team will progress to the next stage of the competition. The team's coach estimates their chances of winning each of their three matches in the league to be 0.6, 0.5 and 0.3 respectively, and believes these probabilities to be independent of each other.

1. Show that the probability of the team winning exactly two of their three matches is 0.36 [4 marks]

Let the random variable \(W\) be the number of matches that the team win in the league.

1. Find the probability distribution of \(W\). [4 marks]
2. Find E\((W)\) and Var\((W)\). [6 marks]
3. Comment on the coach's assumption that the probabilities of success in each of the three matches are independent. [2 marks]

Four letters are taken out of their envelopes for signing. Unfortunately they are replaced randomly, one in each envelope. The probability distribution for the number of letters, \(X\), which are now in the correct envelope is given in the following table.

\(r\)	0	1
P(X = r)	\(\frac{3}{8}\)	\(\frac{1}{3}\)	\(\frac{1}{4}\)	0	\(\frac{1}{24}\)

1. Explain why the case \(X = 3\) is impossible. [1]
2. Explain why P(\(X = 4\)) = \(\frac{1}{24}\). [2]
3. Calculate E(\(X\)) and Var(\(X\)). [5]

Jeremy is a computing consultant who sometimes works at home. The number, \(X\), of days that Jeremy works at home in any given week is modelled by the probability distribution P(\(X = r\)) = \(\frac{1}{40}r(r + 1)\) for \(r = 1, 2, 3, 4\).

1. Verify that P(\(X = 4\)) = \(\frac{1}{2}\). [1]
2. Calculate E(\(X\)) and Var(\(X\)). [5]
3. Jeremy works for 45 weeks each year. Find the expected number of weeks during which he works at home for exactly 2 days. [2]

The number, \(X\), of children per family in a certain city is modelled by the probability distribution P(\(X = r\)) = \(k(6 - r)(1 + r)\) for \(r = 0, 1, 2, 3, 4\).

1. Copy and complete the following table and hence show that the value of \(k\) is \(\frac{1}{50}\). [3]

   \(r\)	0	1	2	3	4
   P(\(X = r\))	\(6k\)	\(10k\)

2. Calculate E(\(X\)). [2]
3. Hence write down the probability that a randomly selected family in this city has more than the mean number of children. [1]

A supermarket chain buys a batch of 10000 scratchcard draw tickets for sale in its stores. 50 of these tickets have a £10 prize, 20 of them have a £100 prize, one of them has a £5000 prize and all of the rest have no prize. This information is summarised in the frequency table below.

Prize money	£0	£10	£100	£5000
Frequency	9929	50	20	1

1. Find the mean and standard deviation of the prize money per ticket. [4]
2. I buy two of these tickets at random. Find the probability that I win either two £10 prizes or two £100 prizes. [3]