5.02e Discrete uniform distribution

2. The discrete random variable \(Q\) has the following probability distribution.

1. Write down the name of this distribution. The discrete random variable \(R\) has the following probability distribution.

   \(r\)	14	24	34	44	54
   \(\mathrm { P } ( R = r )\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 5 }\)

2. State the relationship between \(R\) and \(Q\) in the form \(R = a Q + b\). Given that \(\mathrm { E } ( Q ) = 3\) and \(\operatorname { Var } ( Q ) = 2\),
3. find \(\mathrm { E } ( R )\) and \(\operatorname { Var } ( R )\).

6. In a game two spinners are used. The score on the first spinner is given by the random variable \(A\), which has the following probability distribution:

1. State the name of this distribution.
2. Write down \(\mathrm { E } ( A )\). The score on the second spinner is given by the random variable \(B\), which has the following probability distribution:

   \(b\)	1	2	3
   \(\mathrm { P } ( B = b )\)	\(\frac { 1 } { 2 }\)	\(\frac { 1 } { 4 }\)	\(\frac { 1 } { 4 }\)

3. Find \(\mathrm { E } ( B )\). On each player's turn in the game, both spinners are used and the scores on the two spinners are added together. The total score on the two spinners is given by the random variable \(C\).
4. Show that \(\mathrm { P } ( C = 2 ) = \frac { 1 } { 6 }\).
5. Find the probability distribution of \(C\).
6. Show that \(\mathrm { E } ( C ) = \mathrm { E } ( A ) + \mathrm { E } ( B )\).

3 A fair 8 -sided dice has faces labelled 10, 20, 30, ..., 80 .

1. State the distribution of the score when the dice is rolled once.
2. Write down the probability that, when the dice is rolled once, the score is at least 40 .
3. The dice is rolled three times.
   1. Find the variance of the total score obtained.
   2. Find the probability that on one of the rolls the score is less than 30 , on another it is between 30 and 50 inclusive and on the other it is greater than 50 .

4 A random number generator generates integers between 1 and 50 inclusive, with each number having an equal probability of being generated.

1. State the probability distribution of the numbers generated.
2. Determine the probability that a number generated is within one standard deviation of the mean.

6 An eight-sided dice has its faces numbered \(1,2 , \ldots , 8\). \begin{enumerate}[label=(\alph*)] \item In this part of the question you should assume that the dice is fair.

1. State the probability that, when the dice is rolled once, the score is at least 6 .
2. Show that the probability that the score is within 2 standard deviations of its mean is 1 .

\item A student thinks that the dice may be biased. To investigate this, the student decides to roll the dice 80 times and then carry out a \(\chi ^ { 2 }\) goodness of fit test of a uniform distribution. The spreadsheet below shows the data for the test, where some of the values have been deliberately omitted.

1. Explain why all of the expected frequencies are equal to 10 .
2. Determine the missing values in each of the following cells.
   Carry out the \(\chi ^ { 2 }\) test at the \(5 \%\) significance level.

2 In a game of chance there are 32 slots, numbered 1 to 32, and on each turn a ball lands in one of them. You may assume that the process is completely random. You are given that \(X\) is the random variable denoting the number of the slot that the ball lands in on a given turn.

1. Suggest a suitable distribution to model \(X\). You should state the value(s) of any parameter(s).
2. Write down \(\mathrm { P } ( X = 7 )\). Players of the game start with a score of 0 . On each turn a player may choose to play the game by selecting a number. If the ball lands in the slot with that number then 15 is added to the player's score. Otherwise, the player's score is reduced by 1 . A player's score may become negative. A player decides to play the game, selecting the number 7 on each turn, until the ball lands in the slot numbered 7. You are given that \(Y\) is the random variable denoting the number of turns up to and including the turn in which the ball lands in the slot numbered 7.
3. Determine \(\mathrm { P } ( Y \leqslant 15 )\).
4. Determine the player's expected final score.

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.

5 A fair spinner has five faces, labelled 0, 1, 2, 3, 4.

1. State the distribution of the score when the spinner is spun once.
2. Determine the probability that, when the spinner is spun twice, one of the scores is less than 2 and the other is at least 2.
3. Find the variance of the total score when the spinner is spun 5 times.

2 The discrete random variable \(Y\) is uniformly distributed over the values \(\{ 12,13 , \ldots , 20 \}\).

1. Write down \(\mathrm { P } ( Y < 15 )\).
2. Two independent observations of \(Y\) are taken. Find the probability that one of these values is less than 15 and the other is greater than 15 .
3. Find \(\mathrm { P } ( Y > \mathrm { E } ( Y ) )\).

6 The discrete random variable \(X\) has a uniform distribution over \(\{ n , n + 1 , \ldots , 2 n \}\).

1. Given that \(n\) is odd, find \(\mathrm { P } \left( X < \frac { 3 } { 2 } n \right)\).
2. Given instead that \(n\) is even, find \(\mathrm { P } \left( X < \frac { 3 } { 2 } n \right)\), giving your answer as a single algebraic fraction.
3. The sum of 6 independent values of \(X\) is denoted by \(Y\). Find \(\operatorname { Var } ( Y )\).

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.

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 \(\mathrm { P } \left( \left| \frac { 2 ( X - \mu ) } { \sigma } \right| > 1 \right)\).

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.

6 A lottery has tickets numbered 1 to \(n\) inclusive, where \(n\) is a positive integer. The random variable \(X\) denotes the number on a ticket drawn at random.

1. Determine \(\mathrm { P } \left( \mathrm { X } \leqslant \frac { 1 } { 4 } \mathrm { n } \right)\) in each of the following cases.
   1. \(n\) is a multiple of 4 .
   2. \(n\) is of the form \(4 k + 1\), where \(k\) is a positive integer. Give your answer as a single fraction in terms of \(n\).
2. Given that \(n = 101\), find the probability that \(X\) is within one standard deviation of the mean.

4 A pack of \(k\) cards is labelled \(1,2 , \ldots , k\). A card is drawn at random from the pack. The random variable \(X\) represents the number on the card.

1. Given that \(k > 10\), find \(\mathrm { P } ( X \geqslant 10 )\). You are now given that \(k = 20\).
2. A card is drawn at random from the pack and the number on it is noted. The card is then returned to the pack. This process is repeated until the second occasion on which the number noted is less than 9 . Find the probability that no more than 4 cards have to be drawn. Answer all the questions. Section B (95 marks)

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 \(\mathrm { P } ( W \leqslant 7 )\).

11 The discrete random variable \(X\) has a uniform distribution over the set of all integers between 25 and \(n\) inclusive, where \(n\) is a positive integer with \(n > 25\).

1. Determine \(\mathrm { P } \left( \mathrm { X } < \frac { \mathrm { n } + 25 } { 2 } \right)\) in each of the following cases.

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

1. Given that \(n\) is odd, determine \(\mathrm { P } \left( \mathrm { X } > \frac { 1 } { 2 } \mathrm { n } \right)\), giving your answer as a single fraction in terms of \(n\).
2. Determine the variance of the sum of 10 independent values of \(X\), giving your answer in the form \(\mathrm { an } ^ { 2 } + \mathrm { bn }\), where \(a\) and \(b\) are constants.

1. The continuous random variable X has probability density function

$$f ( x ) = \begin{cases} \frac { 1 } { 8 } & 1 \leqslant x \leqslant 9 \\ 0 & \text { otherwise } \end{cases}$$

1. Write down the name given to this distribution. The continuous random variable \(Y = 5 - 2 X\)
2. Find \(\mathrm { P } ( Y > 0 )\)
3. Find \(\mathrm { E } ( Y )\)
4. Find \(\mathrm { P } ( Y < 0 \mid X < 7.5 )\)

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The random variable \(X\) has a continuous uniform distribution over the interval [5,a], where \(a\) is a constant.
Given that \(\operatorname { Var } ( X ) = \frac { 27 } { 4 }\)

1. show that \(a = 14\) The continuous random variable \(Y\) has probability density function $$f ( y ) = \left\{ \begin{array} { c c } \frac { 1 } { 20 } ( 2 y - 3 ) & 2 \leqslant y \leqslant 6 \\ 0 & \text { otherwise } \end{array} \right.$$ The random variable \(T = 3 \left( X ^ { 2 } + X \right) + 2 Y\)
2. Show that \(\mathrm { E } ( T ) = \frac { 9857 } { 30 }\)

1. The random variable \(X\) has the continuous uniform distribution over the interval [0.5, 2.5]

Talia selects a number, \(T\), at random from the distribution of \(X\)

1. Find \(\mathrm { P } ( T < 1 )\) Malik takes Talia's number, \(T\), and calculates his number, \(M\), where \(M = \frac { 1 } { T ^ { 2 } }\)
2. Find the probability that both \(T\) and \(M\) are less than 2.25 Raja and Greta play a game many times.
   Each time they play they use a number, \(R\), randomly selected from the distribution of \(X\) Raja's score is \(R\) Greta's score is \(G\), where \(G = \frac { 2 } { R ^ { 2 } }\)
3. Determine, giving a reason, who you would expect to have the higher total score.

The random variable \(X\) has a continuous uniform distribution over the interval \([ - 3 , k ]\) Given that \(\mathrm { P } ( - 4 < X < 2 ) = \frac { 1 } { 3 }\)

1. find the value of \(k\) A computer generates a random number, \(Y\), where
   The computer generates 5 random numbers.
2. Calculate the probability that at least 2 of the 5 numbers generated are greater than 7.5

The continuous random variable \(X\) is uniformly distributed over the interval [2, 7]

1. Write down the value of \(\mathrm { E } ( X )\)
2. Find \(\mathrm { P } ( 1 < X < 4 )\)
3. Find \(\mathrm { P } \left( 2 X ^ { 2 } - 15 X + 27 > 0 \right)\)
4. Find \(\mathrm { E } \left( \frac { 3 } { X ^ { 2 } } \right)\)