5.02b Expectation and variance: discrete random variables

4 A fair 8 -sided dice has faces labelled \(1,2 , \ldots , 8\). The random variable \(X\) represents the score when the dice is rolled once.

1. State the distribution of \(X\).
2. Find \(\mathrm { P } ( X < 4 )\).
3. Find each of the following.

1 The random variable \(X\) represents the clutch size (the number of eggs laid) by female birds of a particular species. The probability distribution of \(X\) is given in the table.

1. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   On average 65\% of eggs laid result in a young bird successfully leaving the nest.
2. 1. Find the mean number of young birds that successfully leave the nest.
   2. Find the standard deviation of the number of young birds that successfully leave the nest.

4 The discrete random variable \(X\) has probability distribution defined by $$\mathrm { P } ( X = r ) = k ( 2 r - 1 ) \quad \text { for } r = 1,2,3,4,5,6 \text {, where } k \text { is a constant. }$$

1. Complete the table in the Printed Answer Booklet giving the probabilities in terms of \(k\).

   \(r\)	1	2	3	4	5	6
   \(\mathrm { P } ( X = r )\)

2. Show that the value of \(k\) is \(\frac { 1 } { 36 }\).
3. Draw a graph to illustrate the distribution.
4. In this question you must show detailed reasoning. Find
   A game consists of a player throwing two fair dice. The score is the maximum of the two values showing on the dice.
5. Show that the probability of a score of 3 is \(\frac { 5 } { 36 }\).
6. Show that the probability distribution for the score in the game is the same as the probability distribution of the random variable \(X\).
7. The game is played three times. Find

1 In a game at a charity fair, a spinner is spun 4 times.
On each spin the chance that the spinner lands on a score of 5 is 0.2 .
The random variable \(X\) represents the number of spins on which the spinner lands on a score of 5 .

1. Find \(\mathrm { P } ( X = 3 )\).
2. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   One game costs \(\pounds 1\) to play and, for each spin that lands on a score of 5 , the player receives 50 pence.
3. 1. Find the expected total amount of money gained by a player in one game.
   2. Find the standard deviation of the total amount of money gained by a player in one game.

1 In a quiz a contestant is asked up to four questions. The contestant's turn ends once the contestant gets a question wrong or has answered all four questions. The probability that a particular contestant gets any question correct is 0.6 , independently of other questions. The discrete random variable \(X\) models the number of questions which the contestant gets correct in a turn.

1. Show that \(\mathrm { P } ( X = 4 ) = 0.1296\). The probability distribution of \(X\) is shown in Fig. 1.1. \begin{table}[h]

   \(r\)	0	1	2	3	4
   \(\mathrm { P } ( X = r )\)	0.4	0.24	0.144	0.0864	0.1296

   \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{table}
2. Find each of the following.
   The number of points that a contestant scores is as shown in Fig. 1.2. \begin{table}[h]

   Number of questions correct	Number of points scored
   0 or 1	0
   2	2
   3	3
   4	5

   \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{table} The discrete random variable \(Y\) models the number of points which the contestant scores.
3. Without doing any working, explain whether each of the following will be less than, equal to or greater than the corresponding value for \(X\).

6 The random variable \(X\) has a uniform distribution over the values \(\{ 1,4,7 , \ldots , 3 n - 2 \}\), where \(n\) is a positive integer.

1. Determine \(\operatorname { Var } ( X )\) in terms of \(n\).
2. Given that \(n = 100\), find the probability that \(X\) is within one standard deviation of the mean.

3 A fair four-sided dice has its faces numbered \(0,1,2,3\). The dice is rolled three times. The discrete random variable \(X\) is the sum of the lowest and highest scores obtained.

1. Show that \(\mathrm { P } ( X = 1 ) = \frac { 3 } { 32 }\). The table below shows the probability distribution of \(X\).

   \(r\)	0	1	2	3	4	5	6
   \(\mathrm { P } ( X = r )\)	\(\frac { 1 } { 64 }\)	\(\frac { 3 } { 32 }\)	\(\frac { 13 } { 64 }\)	\(\frac { 3 } { 8 }\)	\(\frac { 13 } { 64 }\)	\(\frac { 3 } { 32 }\)	\(\frac { 1 } { 64 }\)

2. In this question you must show detailed reasoning. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   • The random variable \(Y\) represents the sum of 10 values of \(X\).
     1. State a property of the 10 values of \(X\) that would make it possible to deduce the standard deviation of \(Y\).
     2. Given that this property holds, determine the standard deviation of \(Y\).

7 The discrete random variable \(X\) has a uniform distribution over the set of all integers between 100 and \(n\) inclusive, where \(n\) is a positive integer with \(n > 100\).

1. Given that \(n\) is even, determine \(\mathrm { P } \left( \mathrm { X } < \frac { 100 + \mathrm { n } } { 2 } \right)\).
2. Determine the variance of the sum of 50 independent values of \(X\), giving your answer in the form \(\mathrm { a } \left( \mathrm { n } ^ { 2 } + \mathrm { bn } + \mathrm { c } \right)\), where \(a , b\) and \(c\) are constants.

2 The sides of a fair 12 -sided spinner are labelled \(1,2 , \ldots , 12\). The spinner is spun and \(X\) is the random variable denoting the number on the side of the spinner that it lands on.

1. Suggest a suitable distribution to model \(X\). You should state the value(s) of any parameter(s).
2. Find each of the following.
   You are given that \(\mathrm { E } ( X )\) is denoted by \(\mu\) and \(\operatorname { Var } ( X )\) is denoted by \(\sigma ^ { 2 }\).
3. Determine \(\mathrm { P } \left( \left| \frac { 2 ( X - \mu ) } { \sigma } \right| > 1 \right)\).

6 The probability distribution of a discrete random variable, \(X\), is shown in the table below.

1. Find \(\mathrm { E } ( X )\) in terms of \(a\) and \(b\).
2. 1. In the case where \(\mathrm { E } ( \mathrm { X } ) = \mathrm { a } + 0.4\), find an expression for \(\operatorname { Var } ( X )\) in terms of \(a\).
   2. In this case, show that the greatest possible value of \(\operatorname { Var } ( X )\) is 0.65 . You must state the associated value of \(a\).
3. You are now given instead that \(\mathrm { E } ( X )\) is not known.
   1. State the least possible value of \(\operatorname { Var } ( X )\).
   2. Give all possible pairs of values of \(a\) and \(b\) which give the least possible value of \(\operatorname { Var } ( X )\) stated in part (c)(i).

1 A quiz team of 4 students is to be selected from a group of 7 girls and 5 boys. The team is selected at random from the students in the group. The number of girls in the team is denoted by the random variable \(X\).

1. Show that \(\mathrm { P } ( X = 4 ) = \frac { 7 } { 99 }\). Table 1 shows the probability distribution of \(X\). \begin{table}[h]

   \(r\)	0	1	2	3	4
   \(\mathrm { P } ( X = r )\)	\(\frac { 1 } { 99 }\)	\(\frac { 14 } { 99 }\)	\(\frac { 42 } { 99 }\)	\(\frac { 35 } { 99 }\)	\(\frac { 7 } { 99 }\)

   \captionsetup{labelformat=empty} \caption{Table 1}
   \end{table}
2. Find each of the following.
   It is decided that the quiz team must have at least 1 girl and at least 1 boy, but the team is still otherwise selected at random.
3. Explain whether \(\mathrm { E } ( X )\) would be smaller than, equal to or larger than the value which you found in part (b).

2 On computer monitor screens there are often one or more tiny dots which are permanently dark and do not display any of the image. Such dots are known as 'dead pixels'. Dead pixels occur on screens randomly and independently of each other. A company manufactures three types of monitor, Types A, B and C. For a monitor of Type A, the screen has a total of 2304000 pixels. For this type of monitor, the probability of a randomly chosen pixel being dead is 1 in 500000 . Let \(X\) represent the number of dead pixels on a monitor screen of this type.

1. Explain why you could use either a binomial distribution or a Poisson distribution to model the distribution of \(X\).
2. Use a Poisson distribution to calculate estimates of each of the following probabilities.
   For a monitor of Type B, the probability of a randomly chosen pixel being dead is also 1 in 500 000. The screen of a monitor of Type B has a total of \(n\) pixels. Use a binomial distribution to find the least value of \(n\) for which the probability of finding at least 1 dead pixel is greater than 0.99 . Give your answer in millions correct to 3 significant figures. For a monitor of Type C, the number of dead pixels on the screen is modelled by a Poisson distribution with mean \(\lambda\).
3. Given that the probability of finding at least one dead pixel is 0.8 , find \(\lambda\).

6

1. The random variable \(X\) has a uniform distribution over the values \(\{ 1,2 , \ldots , n \}\). Show that \(\operatorname { Var } ( X )\) is given by \(\frac { 1 } { 12 } \left( n ^ { 2 } - 1 \right)\).
2. The random variable \(Y\) has a uniform distribution over the values \(\{ 1,3,5 , \ldots , 2 n - 1 \}\). Using the result in part (a) or otherwise, show that \(\operatorname { Var } ( Y )\) is given by \(\frac { 1 } { 3 } \left( n ^ { 2 } - 1 \right)\).
3. Given that \(n = 100\), find the least value of \(k\) for which \(\mathrm { P } ( \mu - k \sigma \leqslant Y \leqslant \mu + k \sigma ) = 1\), where the mean and standard deviation of \(Y\) are represented by \(\mu\) and \(\sigma\) respectively.

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

1. Show that \(k = \frac { 1 } { 35 }\). The distribution of \(X\) is shown in the table.

   \(r\)	1	2	3	4	5
   \(\mathrm { P } ( \mathrm { X } = \mathrm { r } )\)	\(\frac { 1 } { 35 }\)	\(\frac { 2 } { 35 }\)	\(\frac { 1 } { 7 }\)	\(\frac { 2 } { 7 }\)	\(\frac { 17 } { 35 }\)

2. Comment briefly on the shape of the distribution.
3. Find each of the following.
   The random variable \(Y\) is given by \(Y = 5 X - 10\).
4. Find each of the following.

1 A fair six-sided dice is rolled three times.
The random variable \(X\) represents the lowest of the three scores.
The probability distribution of \(X\) is given by the formula \(\mathrm { P } ( X = r ) = k \left( 127 - 39 r + 3 r ^ { 2 } \right)\) for \(r = 1,2,3,4,5,6\).

1. Complete the copy of the table in the Printed Answer Booklet.

   \(r\)	1	2	3	4	5	6
   \(\mathrm { P } ( X = r )\)	\(91 k\)	\(61 k\)	\(37 k\)

2. Show that \(k = \frac { 1 } { 216 }\).
3. Draw a graph to illustrate the distribution.
4. Comment briefly on the shape of the distribution.
5. In this question you must show detailed reasoning. Find each of the following.

3 The table shows the probability distribution of the random variable \(X\), where \(a\) and \(b\) are constants.

1. Given that \(\mathrm { E } ( X ) = 1.8\), determine the values of \(a\) and \(b\). The random variable \(Y\) is given by \(Y = 10 - 3 X\).
2. Using the values of \(a\) and \(b\) which you found in part (a), find each of the following.

9 The random variable \(X\) has a discrete uniform distribution over the values \(\{ 0,1,2 , \ldots , 20 \}\).

1. Find \(\mathrm { P } ( X \leqslant 7 )\).
2. Find each of the following.
   The spreadsheet shows a simulation of the distribution of \(X\). Each of the 25 rows of the spreadsheet below the heading row shows a simulation of 10 independent values of \(X\) together with the value of the mean of the 10 values, denoted by \(Y\).

   \includegraphics[max width=\textwidth, alt={}]{77eabbd6-a058-457f-9601-d66f3c2db005-07_38_45_880_279}	A	B	C	D	E	F	G	H	I	J	K	L
   1	\(X _ { 1 }\)	\(X _ { 2 }\)	\(X _ { 3 }\)	\(X _ { 4 }\)	\(X _ { 5 }\)	\(X _ { 6 }\)	\(X _ { 7 }\)	\(X _ { 8 }\)	\(X _ { 9 }\)	\(X _ { 10 }\)	\(Y\)
   2	1	6	2	1	18	6	4	9	11	11	6.9
   3	13	14	12	2	4	11	16	0	16	0	8.8
   4	4	17	1	16	4	10	12	2	18	13	9.7
   5	2	8	12	1	4	16	12	2	15	8	8.0
   6	7	15	16	0	4	7	1	13	0	20	8.3
   7	15	13	10	1	12	0	20	15	16	6	10.8
   8	14	13	17	12	2	18	16	18	9	4	12.3
   9	20	2	12	3	17	3	0	18	15	13	10.3
   10	2	12	5	12	2	6	0	9	10	15	7.3
   11	5	11	13	10	9	17	10	4	20	15	11.4
   12	14	9	9	7	6	20	2	2	11	16	9.6
   13	15	19	18	19	7	6	6	20	3	8	12.1
   14	5	10	6	4	1	19	15	8	17	18	10.3
   15	0	3	15	15	11	12	0	3	9	16	8.4
   16	1	12	1	15	0	4	11	11	9	2	6.6
   17	12	5	0	8	3	8	12	19	13	12	9.2
   18	9	5	1	13	5	4	18	1	1	19	7.6
   19	16	2	20	20	12	17	2	7	8	20	12.4
   20	18	17	3	2	8	18	7	0	11	6	9.0
   21	15	10	7	20	4	0	5	6	11	14	9.2
   22	3	9	10	14	2	1	8	6	0	7	6.0
   23	11	10	11	10	19	11	3	7	10	0	9.2
   24	12	14	6	6	5	20	11	18	10	14	11.6
   25	1	11	5	14	11	10	1	1	2	0	5.6
   26	0	14	7	11	18	5	10	20	11	9	10.5
   27

3. Use the spreadsheet to estimate \(\mathrm { P } ( Y \leqslant 7 )\).
4. Explain why the true value of \(\mathrm { P } ( Y \leqslant 7 )\) is less than \(\mathrm { P } ( X \leqslant 7 )\), relating your answer to \(\operatorname { Var } ( X )\) and \(\operatorname { Var } ( Y )\).
5. The random variable \(W\) is the mean of 30 independent values of \(X\). Determine an estimate of \(\mathrm { P } ( W \leqslant 7 )\).

1 A website simulates the outcome of throwing four fair dice. Ten thousand people take part in a challenge using the website in which they have one attempt at getting four sixes in the four throws of the dice. The number of people who succeed in getting four sixes is denoted by the random variable \(X\).

1. Show that, for each person, the probability that the person gets four sixes is equal to \(\frac { 1 } { 1296 }\).
2. Explain why you could use either a binomial distribution or a Poisson distribution to model the distribution of \(X\).
3. Use a Poisson distribution to calculate each of the following probabilities.
   Determine the probability that no more than 2 people succeed in getting four sixes at least once in their 20 attempts.

11 The random variable \(X\) takes the value 1 with probability \(p\) and the value 0 with probability \(1 - p\).

1. Find each of the following.
   Use the results of part (a) to prove that

1 The number of insurance policy sales made per month by a salesperson is modelled by the random variable \(X\), with probability distribution shown in the table.

1. Find each of the following.
   The salesperson is paid a basic salary of \(\pounds 1000\) per month plus \(\pounds 500\) for each policy that is sold.
2. Find the mean and standard deviation of the salesperson's monthly salary.

1 In a game at a fair, players choose 4 countries from a list of 10 countries. The names of all 10 countries are then put in a box and the player selects 4 of them at random. The random variable \(X\) represents the number of countries that match those which the player originally chose.

1. Show that the probability that a randomly selected player matches all 4 countries is \(\frac { 1 } { 210 }\). Table 1 shows the probability distribution of \(X\). \begin{table}[h]

   \(r\)	0	1	2	3	4
   \(\mathrm { P } ( X = r )\)	\(\frac { 1 } { 14 }\)	\(\frac { 8 } { 21 }\)	\(\frac { 3 } { 7 }\)	\(\frac { 4 } { 35 }\)	\(\frac { 1 } { 210 }\)

   \captionsetup{labelformat=empty} \caption{Table 1}
   \end{table}
2. Find each of the following.
   Find the mean and standard deviation of the player's loss per game.
3. In order to try to attract more customers, the rules will be changed as follows. The game will still cost \(\pounds 1\) to play. The player will get 25 pence back for every country which is matched, plus an additional bonus of \(\pounds 100\) if all four countries are matched. Find the player's mean gain or loss per game with these new rules.

2 On average 1 in 4000 people have a particular antigen in their blood (an antigen is a molecule which may cause an adverse reaction). \begin{enumerate}[label=(\alph*)] \item

1. A random sample of 1200 people is selected. The random variable \(X\) represents the number of people in the sample who have this antigen in their blood. Explain why you could use either a binomial distribution or a Poisson distribution to model the distribution of \(X\).
2. Use either a binomial or a Poisson distribution to calculate each of the following probabilities.

10 The discrete random variables \(X\) and \(Y\) have distributions as follows: \(X \sim \mathrm {~B} ( 20,0.3 )\) and \(Y \sim \operatorname { Po } ( 3 )\). The spreadsheet in Fig. 10 shows a simulation of the distributions of \(X\) and \(Y\). Each of the 20 rows below the heading row consists of a value of \(X\), a value of \(Y\), and the value of \(X - 2 Y\). \begin{table}[h] \end{table}

1. Use the spreadsheet to estimate each of the following.
   The mean of 50 values of \(X - 2 Y\) is denoted by the random variable \(W\).
2. Calculate an estimate of \(\mathrm { P } ( W > 1 )\).

2 In a game at a charity fair, a player rolls 3 unbiased six-sided dice. The random variable \(X\) represents the difference between the highest and lowest scores.

1. Show that \(\mathrm { P } ( X = 0 ) = \frac { 1 } { 36 }\). The table shows the probability distribution of \(X\).

   \(r\)	0	1	2	3	4	5
   \(\mathrm { P } ( \mathrm { X } = \mathrm { r } )\)	\(\frac { 1 } { 36 }\)	\(\frac { 5 } { 36 }\)	\(\frac { 2 } { 9 }\)	\(\frac { 1 } { 4 }\)	\(\frac { 2 } { 9 }\)	\(\frac { 5 } { 36 }\)

2. Draw a graph to illustrate the distribution.
3. Describe the shape of the distribution.
4. In this question you must show detailed reasoning. Find each of the following.
   As a result of playing the game, the player receives \(30 X\) pence from the organiser of the game.
5. Find the variance of the amount that the player receives.
6. The player pays \(k\) pence to play the game. Given that the average profit made by the organiser is 12.5 pence per game, determine the value of \(k\).

3 In air traffic management, air traffic controllers send radio messages to pilots. On receiving a message, the pilot repeats it back to the controller to check that it has been understood correctly. At a particular site, on average \(4 \%\) of messages sent by controllers are not repeated back correctly and so have been misunderstood. You should assume that instances of messages being misunderstood occur randomly and independently.

1. Find the probability that exactly 2 messages are misunderstood in a sequence of 50 messages.
2. Find the probability that in a sequence of messages, the 10th message is the first one which is misunderstood.
3. Find the probability that in a sequence of 20 messages, there are no misunderstood messages.
4. Determine the expected number of messages required for 3 of them to be misunderstood.
5. Determine the probability that in a sequence of messages, the 3rd misunderstood message is the 60th message in the sequence.