5.02a Discrete probability distributions: general

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

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

   \(r\)	1	2	3	4	5
   \(\mathrm { P } ( X = r )\)	\(4 k\)	\(10 k\)

2. Show that the value of \(k\) is 0.01 .
3. Draw a graph to illustrate the distribution.
4. Describe the shape of the distribution.
5. Find each of the following.

1 A fair five-sided spinner has sectors labelled 1, 2, 3, 4, 5. In a game at a stall at a charity event, the spinner is spun twice. The random variable \(X\) represents the lower of the two scores. The probability distribution of \(X\) is given by the formula \(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = \mathrm { k } ( 11 - 2 \mathrm { r } )\) for \(r = 1,2,3,4,5\),
where \(k\) is a constant.

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

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

2. Determine the value of \(k\).
3. Find each of the following.
   Given that the average profit that the stall-holder makes on one game is 25 pence, find the value of \(C\).

1 Ryan has 6 one-pound coins and 4 two-pound coins. Ryan decides to select 3 of these coins at random to donate to a charity. The total value, in pounds, of these 3 coins is denoted by the random variable \(X\).

1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 1 } { 6 }\). The table below shows the probability distribution of \(X\).

   \(r\)	3	4	5	6
   \(\mathrm { P } ( \mathrm { X } = \mathrm { r } )\)	\(\frac { 1 } { 6 }\)	\(\frac { 1 } { 2 }\)	\(\frac { 3 } { 10 }\)	\(\frac { 1 } { 30 }\)

2. Draw a graph to illustrate the distribution.
3. In this question you must show detailed reasoning. Find each of the following.
   Ryan's friend Sasha decides to give the same amount as Ryan does to the charity plus an extra three pounds. The random variable \(Y\) represents the total amount of money, in pounds, given by Ryan and Sasha.
4. Determine each of the following.

1 The probability distribution for a discrete random variable \(X\) is given in the table below.

1. Find the value of \(c\).
2. Find the value of each of the following.
   The random variable \(Y\) is defined by \(Y = 2 X - 3\).
3. Find the value of each of the following.

1 The random variable \(X\) represents the number of cars arriving at a car wash per 10-minute period. From observations over a number of days, an estimate was made of the probability distribution of \(X\). Table 1 shows this estimated probability distribution. \begin{table}[h] \end{table}

1. In this question you must show detailed reasoning. Use Table 1 to calculate estimates of each of the following.
   You should now assume that \(X\) can be modelled by a Poisson distribution with mean equal to the value which you calculated in part (a).
2. 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 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 \(X\).

3 Jane wonders whether the number of wasps entering a wasp's nest per 5 second interval can be modelled by a Poisson distribution with mean \(\mu\). She counts the number of wasps entering the nest over 60 randomly selected 5 -second intervals. The results are shown in Fig. 3.1. \begin{table}[h] \end{table}

1. Show that a suitable estimate for the value of \(\mu\) is 5.1. Fig. 3.2 shows part of a screenshot for a \(\chi ^ { 2 }\) test to assess the goodness of fit of a Poisson model. The sample mean has been used as an estimate for the population mean. Some of the values in the spreadsheet have been deliberately omitted. \begin{table}[h]

   	A	B	C	D	E
   \includegraphics[max width=\textwidth, alt={}]{e8624e9b-5143-49d2-9683-cc3a1082694e-4_132_40_1069_273}	Number of wasps	Observed frequency	Poisson probability	Expected frequency	Chi-squared contribution
   2	\(\leqslant 2\)	7	0.1165	6.9887	0.0000
   3	3	5		8.0874	1.1786
   4	4	12			0.2765
   5	5	10			0.0255
   6	6	10	0.1490	8.9400	0.1257
   7	7	11	0.1086	6.5134	3.0904
   8	\(\geqslant 8\)	5	0.1440	8.6414
   9

   \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{table}
2. Determine the missing values in each of the following cells, giving your answers correct to 4 decimal places.
   Carry out the hypothesis test at the 5\% significance level.
3. Jane also carries out a \(\chi ^ { 2 }\) test for the number of wasps leaving another nest. As part of her calculations, she finds that the probability of no wasps leaving the nest in a 5 -second period is 0.0053 . She finds that a Poisson distribution is also an appropriate model in this case. Find a suitable estimate for the value of the mean number of wasps leaving the nest per 5-second period.

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 \(Y\).

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).

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.

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.

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 \(k\).

1. The probability distribution for the prize money, \(\pounds X\) per ticket, in a local fundraising lottery is shown below.

1. Calculate the value of \(p\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
3. 1. What is the minimum lottery ticket price that the organiser should set in order to make a profit in the long run?
   2. Suggest why, in practice, people would be prepared to pay more than this minimum price.
      

6. Penelope makes 8 cakes per week. Each cake costs \(\pounds 20\) to make and sells for \(\pounds 60\). She always sells at least 5 cakes per week. Any cakes left at the end of the week are donated to a food bank. The probability that 5 cakes are sold in a week is \(0 \cdot 3\). She is twice as likely to sell 6 cakes in a week as she is to sell 7 cakes in a week. The expected profit per week is \(\pounds 206\). Construct a probability distribution for the weekly profit.
Additional page, if required. number Write the question number(s) in the left-hand margin. Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE}

1. A fair six-sided black die has faces numbered \(1,2,2,3,3\) and 4

The random variable \(B\) represents the score when the black die is rolled.

1. Write down the value of \(\mathrm { E } ( B )\) A white die has 6 faces numbered \(1,1,2,4,5\) and \(c\) where \(c > 5\) The discrete random variable \(W\) represents the score when the white die is rolled and has probability distribution given by

   \(w\)	1	2	4	5	\(c\)
   \(\mathrm { P } ( W = w )\)	\(a + b\)	\(a\)	0.3	\(a\)	\(b\)

   Greg and Nilaya play a game with these dice.
   Greg throws the black die and Nilaya throws the white die. Greg wins the game if he scores at least two more than Nilaya, otherwise Greg loses.
   The probability of Greg winning the game is \(\frac { 1 } { 6 }\)
2. Find the value of \(a\) and the value of \(b\) Show your working clearly. The random variable \(X = 2 W - 5\) Given that \(\mathrm { E } ( X ) = 2.6\)
3. find the exact value of \(\operatorname { Var } ( X )\)

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

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

1. Write down, in terms of \(q , \mathrm { P } ( X \leqslant 0 )\)
2. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { 7 } { 15 } + 13 q + 16 r\) Given that \(\mathrm { E } \left( X ^ { 3 } \right) = \mathrm { E } \left( X ^ { 2 } \right) + \mathrm { E } ( 6 X )\)
3. find the value of \(q\) and the value of \(r\)
4. Hence find \(\mathrm { P } \left( X ^ { 3 } > X ^ { 2 } + 6 X \right)\)

1. The probability distribution of the discrete random variable \(X\) is

$$P ( X = x ) = \begin{cases} \frac { k } { x } & \text { for } x = 1,2 \text { and } 3 \\ \frac { m } { 2 x } & \text { for } x = 6 \text { and } 9 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) and \(m\) are positive constants.
Given that \(\mathrm { E } ( X ) = 3.8\), find \(\operatorname { Var } ( X )\)

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

1. Find \(\mathrm { E } ( X )\) Given that \(\operatorname { Var } ( X ) = 8.79\)
2. find \(\mathrm { E } \left( X ^ { 2 } \right)\) The discrete random variable \(Y\) has probability distribution

   \(y\)	- 2	- 1	0	1	2
   \(\mathrm { P } ( Y = y )\)	\(3 a\)	\(a\)	\(b\)	\(a\)	\(c\)

   where \(a\), \(b\) and \(c\) are constants.
   For the random variable \(Y\) $$\mathrm { P } ( Y \leqslant 0 ) = 0.75 \quad \text { and } \quad \mathrm { E } \left( Y ^ { 2 } + 3 \right) = 5$$
3. Find the value of \(a\), the value of \(b\) and the value of \(c\) The random variable \(W = Y - X\) where \(Y\) and \(X\) are independent.
   The random variable \(T = 3 W - 8\)
4. Calculate \(\mathrm { P } ( W > T )\)

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

1. Find in terms of \(q\)
   1. \(\mathrm { E } ( X )\)
   2. \(\mathrm { E } \left( X ^ { 2 } \right)\) Given that \(\operatorname { Var } ( X ) = 3.66\)
2. show that \(q = 0.3\) In a game, the score is given by the discrete random variable \(X\) Given that games are independent,
3. calculate the probability that after the 4th game has been played, the total score is exactly 20 A round consists of 4 games plus 2 bonus games. The bonus games are only played if after the 4th game has been played the total score is exactly 20 A prize of \(\pounds 10\) is awarded if 6 games are played in a round and the total score for the round is at least 27 Bobby plays 3 rounds.
4. Find the probability that Bobby wins at least \(\pounds 10\)

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

where \(r\) and \(k\) are positive constants.
The standard deviation of \(X\) equals the mean of \(X\) Find the exact value of \(r\)