5.02b Expectation and variance: discrete random variables

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

where \(a\) and \(b\) are probabilities.

1. Find \(\mathrm { E } ( X )\) Given that \(\operatorname { Var } ( X ) = 3.9\)
2. find the value of \(a\) and the value of \(b\) The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) each have the same distribution as \(X\)
3. Find \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } > 3 \right)\)

1. In a class experiment, each day for 170 days, a child is chosen at random and spins a large cardboard coin 5 times and the number of heads is recorded.
   The results are summarised in the following table.

Marcus believes that a \(\mathrm { B } ( 5,0.5 )\) distribution can be used to model these data and he calculates expected frequencies, to 2 decimal places, as follows

1. Find the value of \(r\) and the value of \(s\)
2. Carry out a suitable test, at the \(5 \%\) level of significance, to determine whether or not the \(\mathrm { B } ( 5,0.5 )\) distribution is a good model for these data.
   You should state clearly your hypotheses, the test statistic and the critical value used. Nima believes that a better model for these data would be \(\mathrm { B } ( 5 , p )\)
3. Find a suitable estimate for \(p\) To test her model, Nima uses this value of \(p\), to calculate expected frequencies as follows

   Number of heads	0	1	2	3	4	5
   Expected frequency	2.07	14.65	41.44	58.63	41.47	11.74

   The test statistic for Nima's test is 1.62 (to 3 significant figures)
4. State,
   1. giving your reasons, the degrees of freedom
   2. the critical value
      that Nima should use for a test at the 5\% significance level.
5. With reference to Marcus' and Nima's test results, comment on
   1. the probability of the coin landing on heads,
   2. the independence of the spins of the coin. Give reasons for your answers.

1. The discrete random variable \(X\) has probability generating function

$$\mathrm { G } _ { X } ( t ) = \frac { t ^ { 2 } } { ( 3 - 2 t ) ^ { 2 } }$$

1. Specify the distribution of \(X\) A fair die is rolled repeatedly.
2. Describe an outcome that could be modelled by the random variable \(X\)
3. Use calculus and \(\mathrm { G } _ { X } ( t )\) to find
   1. \(\mathrm { E } ( X )\)
   2. \(\operatorname { Var } ( X )\) The discrete random variable \(Y\) has probability generating function $$\mathrm { G } _ { Y } ( t ) = \frac { t ^ { 10 } } { \left( 3 - 2 t ^ { 3 } \right) ^ { 2 } }$$
   (d) Find the exact value of \(\mathrm { P } ( Y = 19 )\)

Each time a spinner is spun, the probability that it lands on red is 0.2

1. Find the probability that the spinner lands on red
   1. for the 1st time on the 4th spin
   2. for the 3rd time on the 8th spin
   3. exactly 4 times during 10 spins Each time the spinner is spun, the probability that it lands on yellow is 0.4
      In a game with this spinner, a player must choose one of two events \(R\) is the event that the spinner lands on red for the \(\mathbf { 1 s t }\) time in at most 4 spins \(Y\) is the event that the spinner lands on yellow for the 3rd time in at most 7 spins
2. Showing your calculations clearly, determine which of these events has the greater probability.

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

1. Find \(\operatorname { Var } ( X )\)
2. Find \(\operatorname { Var } \left( X ^ { 2 } \right)\)

Some of the components produced by a factory are defective. The management requires that no more than \(3 \%\) of the components produced are defective.
Niluki monitors the production process and takes a random sample of \(n\) components.

1. Write down the hypotheses Niluki should use in a test to assess whether or not the proportion of defective components is greater than 0.03 Niluki defines the random variable \(D _ { n }\) to represent the number of defective components in a sample of size \(n\). She considers two tests \(\mathbf { A }\) and \(\mathbf { B }\) In test \(\mathbf { A }\), Niluki uses \(n = 100\) and if \(D _ { 100 } \geqslant 5\) she rejects \(H _ { 0 }\)
2. Find the size of test \(\mathbf { A }\) In test B, Niluki uses \(n = 80\) and
   • if \(D _ { 80 } \geqslant 5\) she rejects \(\mathrm { H } _ { 0 }\)
   • if \(D _ { 80 } \leqslant 3\) she does not reject \(\mathrm { H } _ { 0 }\)
   • if \(D _ { 80 } = 4\) she takes a second random sample of size 80 and if \(D _ { 80 } \geqslant 1\) in this second sample then she rejects \(\mathrm { H } _ { 0 }\) otherwise she does not reject \(\mathrm { H } _ { 0 }\)
   • Find the size of test \(\mathbf { B }\)
   Given that the actual proportion of defective components is 0.06
3. 1. find the power of test \(\mathbf { A }\)
   2. find the expected number of components sampled using test \(\mathbf { B }\) Given also that, when the actual proportion of defective components is 0.06 , the power of test \(\mathbf { B }\) is 0.713
4. suggest, giving your reasons, which test Niluki should use.

1. The random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( t )\) where

$$\mathrm { G } _ { X } ( t ) = \frac { 1 } { \sqrt { 4 - 3 t } }$$

1. Use calculus to find \(\operatorname { Var } ( X )\) Show your working clearly.
2. Find the exact value of \(\mathrm { P } ( X \leqslant 2 )\) The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) each have the same distribution as \(X\) The random variable \(Y = X _ { 1 } + X _ { 2 } + 1\)
3. By finding the probability generating function of \(Y\), state the name of the distribution of \(Y\)
4. Hence, or otherwise, find \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } > 5 \right)\)

1. The probability of winning a prize when playing a single game of Pento is \(\frac { 1 } { 5 }\)

When more than one game is played the games are independent.
Sam plays 20 games.

1. Find the probability that Sam wins 4 or more prizes. Tessa plays a series of games.
2. Find the probability that Tessa wins her 4th prize on her 20th game. Rama invites Sam and Tessa to play some new games of Pento. They must pay Rama \(\pounds 1\) for each game they play but Rama will pay them \(\pounds 2\) for the first time they win a prize, \(\pounds 4\) for the second time and \(\pounds ( 2 w )\) when they win their \(w\) th prize ( \(w > 2\) ) Sam decides to play \(n\) games of Pento with Rama.
3. Show that Sam's expected profit is \(\pounds \frac { 1 } { 25 } \left( n ^ { 2 } - 16 n \right)\) Given that Sam chose \(n = 15\)
4. find the probability that Sam does not make a loss. Tessa agrees to play Pento with Rama. She will play games until she wins \(r\) prizes and then she will stop.
5. Find, in terms of \(r\), Tessa's expected profit.

1. A call centre routes incoming telephone calls to agents who have specialist knowledge to deal with the call. The probability of a caller, chosen at random, being connected to the wrong agent is p.

The probability of at least 1 call in 5 consecutive calls being connected to the wrong agent is 0.049 The call centre receives 1000 calls each day.

1. Find the mean and variance of the number of wrongly connected calls a day.
2. Use a Poisson approximation to find, to 3 decimal places, the probability that more than 6 calls each day are connected to the wrong agent.
3. Explain why the approximation used in part (b) is valid. The probability that more than 6 calls each day are connected to the wrong agent using the binomial distribution is 0.8711 to 4 decimal places.
4. Comment on the accuracy of your answer in part (b).
   

1. Bags of \(\pounds 1\) coins are paid into a bank. Each bag contains 20 coins.

The bank manager believes that \(5 \%\) of the \(\pounds 1\) coins paid into the bank are fakes. He decides to use the distribution \(X \sim B ( 20,0.05 )\) to model the random variable \(X\), the number of fake \(\pounds 1\) coins in each bag. The bank manager checks a random sample of 150 bags of \(\pounds 1\) coins and records the number of fake coins found in each bag. His results are summarised in Table 1. He then calculates some of the expected frequencies, correct to 1 decimal place. \begin{table}[h] \end{table}

1. Carry out a hypothesis test, at the \(5 \%\) significance level, to see if the data supports the bank manager's statistical model. State your hypotheses clearly. The assistant manager thinks that a binomial distribution is a good model but suggests that the proportion of fake coins is higher than \(5 \%\). She calculates the actual proportion of fake coins in the sample and uses this value to carry out a new hypothesis test on the data. Her expected frequencies are shown in Table 2. \begin{table}[h]

   Number of fake coins in each bag	0	1	2	3	4 or more
   Observed frequency	43	62	26	13	6
   Expected frequency	44.5	55.7	33.2	12.5	4.1

   \captionsetup{labelformat=empty} \caption{Table 2}
   \end{table}
2. Explain why there are 2 degrees of freedom in this case.
3. Given that she obtains a \(\chi ^ { 2 }\) test statistic of 2.67 , test the assistant manager's hypothesis that the binomial distribution is a good model for the number of fake coins in each bag. Use a \(5 \%\) level of significance and state your hypotheses clearly.
   

1. The probability generating function of the discrete random variable \(X\) is given by

$$G _ { x } ( t ) = k \left( 3 + t + 2 t ^ { 2 } \right) ^ { 2 }$$

1. Show that \(\mathrm { k } = \frac { 1 } { 36 }\)
2. Find \(\mathrm { P } ( \mathrm { X } = 3 )\)
3. Show that \(\operatorname { Var } ( \mathrm { X } ) = \frac { 29 } { 18 }\)
4. Find the probability generating function of \(2 \mathrm { X } + 1\) \section*{Q uestion 6 continued} \section*{Q uestion 6 continued} \section*{Q uestion 6 continued}

1. Sam and Tessa are testing a spinner to see if the probability, p , of it landing on red is less than \(\frac { 1 } { 5 }\). They both use a \(10 \%\) significance level.

Sam decides to spin the spinner 20 times and record the number of times it lands on red.

1. Find the critical region for Sam's test.
2. Write down the size of Sam's test. Tessa decides to spin the spinner until it lands on red and she records the number of spins.
3. Find the critical region for Tessa's test.
4. Find the size of Tessa's test.
5. 1. Show that the power function for Sam's test is given by $$( 1 - p ) ^ { 19 } ( 1 + 19 p )$$
   2. Find the power function for Tessa's test.
6. With reference to parts (b), (d) and (e), state, giving your reasons, whether you would recommend Sam's test or Tessa's test when \(\mathrm { p } = 0.15\)

1. Korhan and Louise challenge each other to find an estimator for the mean, \(\mu\), of the continuous random variable \(X\) which has variance \(\sigma ^ { 2 }\) \(X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { n }\) are \(n\) independent observations taken from \(X\) Korhan's estimator is given by

$$K = \frac { 2 } { n ( n + 1 ) } \sum _ { r = 1 } ^ { n } r X _ { r }$$ Louise's estimator is given by $$L = \frac { X _ { 1 } + X _ { 2 } } { 3 } + \frac { X _ { 3 } + X _ { 4 } + \ldots + X _ { n } } { 3 ( n - 2 ) }$$

1. Show that \(K\) and \(L\) are both unbiased estimators of \(\mu\)
2. 1. Find \(\operatorname { Var } ( K )\)
   2. Find \(\operatorname { Var } ( L )\) The winner of the challenge is the person who finds the better estimator.
3. Determine the winner of the challenge for large values of \(n\). Give reasons for your answer.

1. A bag contains a large number of marbles of which an unknown proportion, \(p\), is yellow.

Three random samples of size \(n\) are taken, and the number of yellow marbles in each sample, \(Y _ { 1 } , Y _ { 2 }\) and \(Y _ { 3 }\), is recorded. Two estimators \(\hat { \mathrm { p } } _ { 1 }\) and \(\hat { \mathrm { p } } _ { 2 }\) are proposed to estimate the value of \(p\) $$\begin{aligned} & \hat { p } _ { 1 } = \frac { Y _ { 1 } + 3 Y _ { 2 } - 2 Y _ { 3 } } { 2 n } \\ & \hat { p } _ { 2 } = \frac { 2 Y _ { 1 } + 3 Y _ { 2 } + Y _ { 3 } } { 6 n } \end{aligned}$$

1. Show that \(\hat { \mathrm { p } } _ { 1 }\) and \(\hat { \mathrm { p } } _ { 2 }\) are both unbiased estimators of \(p\)
2. Find the variance of \(\hat { p } _ { 1 }\) The variance of \(\hat { \mathrm { p } } _ { 2 }\) is \(\frac { 7 p ( 1 - p ) } { 18 n }\)
3. State, giving a reason, which is the better estimator. The estimator \(\hat { p } _ { 3 } = \frac { Y _ { 1 } + a Y _ { 2 } + 3 Y _ { 3 } } { b n }\) where \(a\) and \(b\) are positive integers.
4. Find the pair of values of \(a\) and \(b\) such that \(\hat { \mathrm { p } } _ { 3 }\) is a better unbiased estimator of \(p\) than both \(\hat { \mathrm { p } } _ { 1 }\) and \(\hat { \mathrm { p } } _ { 2 }\) You must show all stages of your working.

3 In this question you must show detailed reasoning. A discrete random variable \(V\) has the following probability distribution, where \(p\) and \(q\) are constants. It is given that \(\mathrm { E } ( V ) = \operatorname { Var } ( V )\). Determine the value of \(p\) and the value of \(q\).

5 A spinner has 5 edges. Each edge is numbered with a different integer from 1 to 5 . When the spinner is spun, it is equally likely to come to rest on any one of the edges. The spinner is spun 100 times. The number of times on which the spinner comes to rest on the edge numbered 5 is denoted by \(X\).

1. \(\mathrm { E } ( X )\),
2. \(\operatorname { Var } ( X )\).
   1. Write down
   2. Use a normal distribution with the same mean and variance as in your answers to part (i) to estimate the smallest value of \(n\) such that \(\mathrm { P } ( X \geqslant n ) < 0.02\).
   3. Use the binomial distribution to find exactly the smallest value of \(n\) such that \(\mathrm { P } ( X \geqslant n ) < 0.02\). Show the values of all relevant calculations.

4 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 \(\mathrm { E } ( N ) = 2.55\).

1. Find \(\operatorname { Var } ( N )\).
2. Find \(\mathrm { E } ( 3 N + 2 )\).
3. Find \(\operatorname { Var } ( 3 N + 2 )\).

2 In a fairground game a competitor scores \(0,1,2\) or 3 with probabilities given in the following table, where \(a\) and \(b\) are constants. The competitor's expected score is 0.9 .

1. Show that \(b = 0.15\).
2. Find the variance of the score.
3. The competitor has to pay \(\pounds 2.50\) to take part, and wins a prize of \(\pounds 2 X\), where \(X\) is the score achieved. Find the expectation of the competitor's loss.

1. The random variable \(W\) has a discrete uniform distribution where

$$\mathrm { P } ( W = w ) = \frac { 1 } { 5 } \quad \text { for } w = 1,2,3,4,5$$

1. Find \(\mathrm { P } ( 2 \leqslant W < 3.5 )\) The discrete random variable \(X = 5 - 2 W\)
2. Find \(\mathrm { E } ( X )\)
3. Find \(\mathrm { P } ( X < W )\) The discrete random variable \(\mathrm { Y } = \frac { 1 } { W }\)
4. Find
   1. the probability distribution of \(Y\)
   2. \(\operatorname { Var } ( Y )\), showing your working.
5. Find \(\operatorname { Var } ( 2 - 3 Y )\)

1. A biased coin has probability 0.4 of showing a head. In an experiment, the coin is spun until a head appears. If a head has not appeared after 4 spins, the coin is not spun again. The random variable \(X\) represents the number of times the coin is spun.

For example, \(X = 3\) if the first two spins do not show a head but the third spin does show a head. The coin would not then be spun a fourth time since the coin has already shown a head.

1. Show that \(\mathrm { P } ( X = 3 ) = 0.144\) The table gives some values for the probability distribution of \(X\)

   \(x\)	1	2	3	4
   \(\mathrm { P } ( X = x )\)		0.24	0.144

2. 1. Write down the value of \(\mathrm { P } ( X = 1 )\)
   2. Find \(\mathrm { P } ( X = 4 )\)
3. Find \(\mathrm { E } ( X )\)
4. Find \(\operatorname { Var } ( X )\) The random variable \(H\) represents the number of heads obtained when the coin is spun in the experiment.
5. Explain why \(H\) can only take the values 0 and 1 and find the probability distribution of \(H\).
6. Write down the value of
   1. \(\mathrm { P } ( \{ X = 3 \} \cap \{ H = 0 \} )\)
   2. \(\mathrm { P } ( \{ X = 4 \} \cap \{ H = 0 \} )\) The random variable \(S = X + H\)
7. Find the probability distribution of \(S\)

1. The discrete random variable \(D\) with the following probability distribution represents the score when a 4-sided die is rolled.

1. Write down the name of this distribution. The die is used to play a game and the random variable \(X\) represents the number of points scored. The die is rolled once and if \(D = 2,3\) or 4 then \(X = D\). If \(D = 1\) the die is rolled a second time and \(X = 0\) if \(D = 1\) again, otherwise \(X\) is the sum of the two scores on the die.
2. Show that the probability of scoring 3 points in this game is \(\frac { 5 } { 16 }\)
3. Find the probability of scoring 0 in this game. The table below shows the probability distribution for the remaining values of \(X\).

   \(x\)	0	2	3	4	5
   \(\mathrm { P } ( X = x )\)		\(\frac { 1 } { 4 }\)		\(\frac { 5 } { 16 }\)	\(\frac { 1 } { 16 }\)

4. Find \(\mathrm { E } ( X )\)
5. Find \(\operatorname { Var } ( X )\) The discrete random variable \(R\) represents the number of times the die is rolled in the game.
6. Write down the probability distribution of \(R\). The random variable \(Y = 2 R + 0.5\)
7. Show that \(\mathrm { E } ( Y ) = \mathrm { E } ( X )\) The game is played once.
8. Find \(\mathrm { P } ( X > Y )\)

4. Three bags A, B and \(\mathbf { C }\) each contain coloured balls. Bag A contains 4 red balls and 2 yellow balls only.
Bag B contains 4 red balls and 1 yellow ball only.
Bag \(\mathbf { C }\) contains 6 red balls only. In a game
Mike takes a ball at random from bag \(\mathbf { A }\), records the colour and places it in bag \(\mathbf { C }\). He then takes a ball at random from bag \(\mathbf { B }\), records the colour and places it in bag \(\mathbf { C }\). Finally, Mike takes a ball at random from bag \(\mathbf { C }\) and records the colour.

1. Complete the tree diagram on the page opposite, to illustrate the game by adding the remaining branches and all probabilities.
2. Show that the probability that Mike records a yellow ball exactly twice is \(\frac { 1 } { 10 }\) Given that Mike records exactly 2 yellow balls,
3. find the probability that the ball drawn from bag \(\mathbf { A }\) is red. Mike plays this game a large number of times, each time starting with the bags containing balls as described above. The random variable \(X\) represents the number of yellow balls recorded in a single game.
4. Find the probability distribution of \(X\)
5. Find \(\mathrm { E } ( X )\) Bag B
   Bag C \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Bag A} \includegraphics[alt={},max width=\textwidth]{29ac0c0b-f963-40a1-beba-7146bbb2d021-13_739_1580_411_182}
   \end{figure}

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

where \(q , r\) and \(u\) are probabilities.

1. Write down the value of \(\mathrm { E } ( Y )\) The cumulative distribution function of \(Y\) is \(\mathrm { F } ( y )\) Given that \(F ( 0 ) = \frac { 19 } { 30 }\)
2. show that the value of \(u\) is \(\frac { 4 } { 15 }\) Given also that \(\operatorname { Var } ( Y ) = 37\)
3. find the value of \(q\) and the value of \(r\) The coordinates of a point \(P\) are \(( 12 , Y )\) The random variable \(D\) represents the length of \(O P\)
4. Find the probability distribution of \(D\)

7. A music teacher monitored the sight-reading ability of one of her pupils over a 10 week period. At the end of each week, the pupil was given a new piece to sight-read and the teacher noted the number of errors \(y\). She also recorded the number of hours \(x\) that the pupil had practised each week. The data are shown in the table below.

1. Given that \(\mathrm { E } ( X ) = - 0.2\), find the value of \(\alpha\) and the value of \(\beta\).
2. Write down \(\mathrm { F } ( 0.8 )\).
   1. Evaluate \(\operatorname { Var } ( X )\).

4. The discrete random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = \begin{array} { l l } k \left( x ^ { 2 } - 9 \right) , & x = 4,5,6 \\ 0 , & \text { otherwise } \end{array}$$ where \(k\) is a positive constant.

1. Show that \(k = \frac { 1 } { 50 }\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
3. Find \(\operatorname { Var } ( 2 X - 3 )\).