5.02a Discrete probability distributions: general

4 A fair die with faces numbered \(1,2,2,2,3,6\) is thrown. The score, \(X\), is found by squaring the number on the face the die shows and then subtracting 4.

1. Draw up a table to show the probability distribution of \(X\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

2 A random variable \(X\) has the probability distribution shown in the following table, where \(p\) is a constant.

1. Find the value of \(p\).
2. Given that \(\mathrm { E } ( X ) = 1.15\), find \(\operatorname { Var } ( X )\).

6 A fair red spinner has 4 sides, numbered 1,2,3,4. A fair blue spinner has 3 sides, numbered 1,2,3. When a spinner is spun, the score is the number on the side on which it lands. The spinners are spun at the same time. The random variable \(X\) denotes the score on the red spinner minus the score on the blue spinner.

1. Draw up the probability distribution table for \(X\).
2. Find \(\operatorname { Var } ( X )\).
3. Find the probability that \(X\) is equal to 1 , given that \(X\) is non-zero.

2 A fair 6 -sided die has the numbers \(- 1 , - 1,0,0,1,2\) on its faces. A fair 3 -sided spinner has edges numbered \(- 1,0,1\). The die is thrown and the spinner is spun. The number on the uppermost face of the die and the number on the edge on which the spinner comes to rest are noted. The sum of these two numbers is denoted by \(X\).

1. Draw up a table showing the probability distribution of \(X\).
2. Find \(\operatorname { Var } ( X )\).

4 In a probability distribution the random variable \(X\) takes the values \(- 1,0,1,2,4\). The probability distribution table for \(X\) is as follows.

1. Find the value of \(p\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
3. Given that \(X\) is greater than zero, find the probability that \(X\) is equal to 2 .

4 A fair spinner has five sides numbered \(1,2,3,4,5\). The score on one spin is denoted by \(X\).

1. Show that \(\operatorname { Var } ( X ) = 2\).
   Fiona has another spinner, also with five sides numbered \(1,2,3,4,5\). She suspects that it is biased so that the expected score is less than 3 . In order to test her suspicion, she plans to spin her spinner 40 times. If the mean score is less than 2.6 she will conclude that her spinner is biased in this way.
2. Find the probability of a Type I error.
3. State what is meant by a Type II error in this context.

4 The score on one spin of a 5 -sided spinner is denoted by the random variable \(X\) with probability distribution as shown in the table.

1. Show that \(\operatorname { Var } ( X ) = 1.2\).
   The spinner is spun 200 times. The score on each spin is noted and the mean, \(\bar { X }\), of the 200 scores is found.
2. Given that \(\mathrm { P } ( \bar { X } > a ) = 0.1\), find the value of \(a\).
3. Explain whether it was necessary to use the Central Limit theorem in your answer to part (b).
4. Johann has another, similar, spinner. He suspects that it is biased so that the mean score is less than 2 . He spins his spinner 200 times and finds that the mean of the 200 scores is 1.86 . Given that the variance of the score on one spin of this spinner is also 1.2 , test Johann's suspicion at the 5\% significance level.

1 A fair coin is tossed 5 times and the number of heads is recorded.

1. The random variable \(X\) is the number of heads. State the mean and variance of \(X\).
2. The number of heads is doubled and denoted by the random variable \(Y\). State the mean and variance of \(Y\).

1 Ashok has 3 green pens and 7 red pens. His friend Rod takes 3 of these pens at random, without replacement. Draw up a probability distribution table for the number of green pens Rod takes.

4 The discrete random variable \(X\) has probability generating function \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\) given by $$G _ { X } ( t ) = 0.2 t + 0.5 t ^ { 2 } + 0.3 t ^ { 3 }$$ The random variable \(Y\) is the sum of two independent observations of \(X\).

1. Find the probability generating function of \(Y\), giving your answer as an expanded polynomial in \(t\). [3]
2. Use the probability generating function of \(Y\) to find \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\).

6 Tanji has a bag containing 4 red balls and 2 blue balls. He selects 3 balls at random from the bag, without replacement. The number of red balls selected by Tanji is denoted by \(X\).

1. Find the probability generating function \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\) of \(X\).
   Tanji also has two coins, each biased so that the probability of obtaining a head when it is thrown is \(\frac { 1 } { 4 }\). He throws the two coins at the same time. The number of heads obtained is denoted by \(Y\).
2. Find the probability generating function \(\mathrm { G } _ { Y } ( \mathrm { t } )\) of \(Y\).
   The random variable \(Z\) is the sum of the number of red balls selected by Tanji and the number of heads obtained.
3. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
4. Use the probability generating function of \(Z\) to find \(E ( Z )\) and \(\operatorname { Var } ( Z )\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 A bag contains 4 red balls and 6 blue balls. Rassa selects two balls at random, without replacement, from the bag. The number of red balls selected by Rassa is denoted by \(X\).

1. Find the probability generating function, \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\), of \(X\).
   Rassa also tosses two coins. One coin is biased so that the probability of a head is \(\frac { 2 } { 3 }\). The other coin is biased so that the probability of a head is \(p\). The probability generating function of \(Y\), the number of heads obtained by Rassa, is \(\mathrm { G } _ { Y } ( \mathrm { t } )\). The coefficient of \(t\) in \(\mathrm { G } _ { Y } ( \mathrm { t } )\) is \(\frac { 7 } { 12 }\).
2. Find \(\mathrm { G } _ { Y } ( \mathrm { t } )\).
   The random variable \(Z\) is the sum of the number of red balls selected and the number of heads obtained by Rassa.
3. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
4. Use the probability generating function of \(Z\) to find \(\mathrm { E } ( Z )\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 A fair six-sided die has faces numbered \(1,2,3,4,5,6\). The score on one throw is denoted by \(X\).

1. Write down the value of \(\mathrm { E } ( X )\) and show that \(\operatorname { Var } ( X ) = \frac { 35 } { 12 }\). Fayez has a six-sided die with faces numbered \(1,2,3,4,5,6\). He suspects that it is biased so that when it is thrown it is more likely to show a low number than a high number. In order to test his suspicion, he plans to throw the die 50 times. If the mean score is less than 3 he will conclude that the die is biased.
2. Find the probability of a Type I error.
3. With reference to this context, describe circumstances in which Fayez would make a Type II error.

5 Harry has three coins.

• One coin is biased so that, when it is thrown, the probability of obtaining a head is \(\frac { 1 } { 3 }\).
• The second coin is biased so that, when it is thrown, the probability of obtaining a head is \(\frac { 1 } { 4 }\).
• The third coin is biased so that, when it is thrown, the probability of obtaining a head is \(\frac { 1 } { 5 }\).

The random variable \(X\) is the number of heads that Harry obtains when he throws all three coins together.

1. Find the probability generating function of \(X\).
   Isaac has two fair coins. The random variable \(Y\) is the number of heads that Isaac obtains when he throws both of his coins together. The random variable \(Z\) is the total number of heads obtained when Harry throws his three coins and Isaac throws his two coins.
2. Find the probability generating function of \(Z\), expressing your answer as a polynomial in \(t\).
3. Use the probability generating function of \(Z\) to find \(E ( Z )\).

5 The random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( \mathrm { t } )\) given by $$\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } ) = \mathrm { k } \left( 1 + 3 \mathrm { t } + 4 \mathrm { t } ^ { 2 } \right)$$ where \(k\) is a constant.

1. Show that \(\mathrm { E } ( X ) = \frac { 11 } { 8 }\).
   The random variable \(Y\) has probability generating function \(\mathrm { G } _ { \gamma } ( \mathrm { t } )\) given by $$G _ { \gamma } ( t ) = \frac { 1 } { 3 } t ^ { 2 } ( 1 + 2 t )$$ The random variables \(X\) and \(Y\) are independent and \(\mathrm { Z } = \mathrm { X } + \mathrm { Y }\).
2. Find the probability generating function of \(Z\), expressing your answer as a polynomial in \(t\).
3. Use your answer to part (b) to find the value of \(\operatorname { Var } ( Z )\).
4. Write down the most probable value of \(Z\).

4 The random variable \(Y\) is the sum of two independent observations of the random variable \(X\). The probability generating function \(\mathrm { G } _ { Y } ( \mathrm { t } )\) of \(Y\) is given by $$G _ { Y } ( t ) = \frac { t ^ { 2 } } { ( 4 - 3 t ) ^ { 4 } }$$

1. Find \(\mathrm { E } ( \mathrm { Y } )\).
2. Write down an expression for the probability generating function of \(X\).
3. Find \(\mathrm { P } ( X = 4 )\).

4 The random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( t )\) given by $$\mathrm { G } _ { X } ( t ) = \operatorname { ct } ( 1 + t ) ^ { 5 }$$ where \(c\) is a constant.

1. Find the value of \(c\).
2. Find the value of \(\mathrm { E } ( X )\). \includegraphics[max width=\textwidth, alt={}, center]{b5ff998a-fcb6-4a1b-ae86-ec66b0dccc3c-06_2718_33_141_2014} The random variable \(Y\) is the sum of two independent values of \(X\).
3. Write down the probability generating function of \(Y\) and hence find \(\operatorname { Var } ( Y )\).
4. Find \(\mathrm { P } ( Y = 5 )\).

5 Keira has two unbiased coins. She tosses both coins. The number of heads obtained by Keira is denoted by \(X\).

1. Find the probability generating function \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\) of \(X\).
   Hassan has three coins, two of which are biased so that the probability of obtaining a head when the coin is tossed is \(\frac { 1 } { 3 }\). The corresponding probability for the third coin is \(\frac { 1 } { 4 }\). The number of heads obtained by Hassan when he tosses these three coins is denoted by \(Y\).
2. Find the probability generating function \(\mathrm { G } _ { Y } ( \mathrm { t } )\) of \(Y\).
   The random variable \(Z\) is the total number of heads obtained by Keira and Hassan.
3. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
4. Use the probability generating function of \(Z\) to find \(\mathrm { E } ( Z )\).
5. Use the probability generating function of \(Z\) to find the most probable value of \(Z\).

5 The random variable \(X\) has the binomial distribution \(\mathrm { B } ( n , p )\).

1. Write down an expression for \(\mathrm { P } ( \mathrm { X } = \mathrm { r } )\) and hence show that the probability generating function of \(X\) is \(( \mathrm { q } + \mathrm { pt } ) ^ { \mathrm { n } }\), where \(\mathrm { q } = 1 - \mathrm { p }\).
2. Use the probability generating function of \(X\) to prove that \(\mathrm { E } ( \mathrm { X } ) = \mathrm { np }\) and \(\operatorname { Var } ( \mathrm { X } ) = \mathrm { np } ( 1 - \mathrm { p } )\). [5]

5 Nine balls labelled \(1,2,3,4,5,6,7,8,9\) are placed in a bag. Kai selects three balls at random from the bag, without replacement. The random variable \(X\) is the number of balls selected by Kai that are labelled with a multiple of 3 .

1. Find the probability generating function \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\) of \(X\).
   The balls are replaced in the bag.
   Jacob now selects two balls at random from the bag, without replacement. The random variable \(Y\) is the number of balls selected by Jacob that are labelled with an even number.
2. Find the probability generating function \(\mathrm { G } _ { Y } ( \mathrm { t } )\) of \(Y\).
   The random variable \(Z\) is the sum of the number of balls that are labelled with a multiple of 3 selected by Kai and the number of balls that are labelled with an even number selected by Jacob.
3. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
4. Use the probability generating function of \(Z\) to find \(\mathrm { E } ( Z )\).

5 The random variable \(X\) is such that \(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = \mathrm { kr } ^ { 2 }\) for \(r = 1,2,3,4\), where \(k\) is a constant.

1. Find the value of \(k\).
2. Find the probability generating function \(\mathrm { G } _ { X } ( \mathrm { t } )\) of \(X\).
   The random variable \(Y\) has probability generating function \(\mathrm { G } _ { Y } ( \mathrm { t } ) = \frac { 1 } { 4 } + \frac { 1 } { 2 } \mathrm { t } + \frac { 1 } { 4 } \mathrm { t } ^ { 2 }\).
   The random variable \(Z\) is the sum of \(X\) and \(Y\).
3. Assuming that \(X\) and \(Y\) are independent, find the probability generating function \(\mathrm { G } _ { \mathrm { Z } } ( \mathrm { t } )\) of \(Z\) as a polynomial in \(t\).
4. Given that \(\mathrm { E } ( \mathrm { Z } ) = \frac { 13 } { 3 }\), use \(\mathrm { G } _ { \mathrm { Z } } ( \mathrm { t } )\) to find \(\operatorname { Var } ( \mathrm { Z } )\).

4 Jason has three biased coins. For each coin the probability of obtaining a head when it is thrown is \(\frac { 2 } { 3 }\). Jason throws all three coins. The number of heads obtained is denoted by \(X\).

1. Find the probability generating function \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\) of \(X\).
   Jason also has two unbiased coins. He throws all five coins. The number of heads obtained from the two unbiased coins is denoted by \(Y\). It is given that \(G _ { Y } ( t ) = \frac { 1 } { 4 } + \frac { 1 } { 2 } t + \frac { 1 } { 4 } t ^ { 2 }\). The random variable \(Z\) is the total number of heads obtained when Jason throws all five coins.
2. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
3. Find \(\mathrm { E } ( \mathrm { Z } )\).

5 A 6 -sided dice, \(A\), with faces numbered \(1,2,3,4,5,6\) is biased so that the probability of throwing a 6 is \(\frac { 1 } { 4 }\). The random variable \(X\) is the number of 6s obtained when dice \(A\) is thrown twice.

1. Find the probability generating function of \(X\).
   A second dice, \(B\), with faces numbered \(1,2,3,4,5,6\) is unbiased. The random variable \(Y\) is the number of 6s obtained when dice \(B\) is thrown twice. The random variable \(Z\) is the total number of 6s obtained when both dice are thrown twice.
2. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
3. Find \(\operatorname { Var } ( Z )\).
4. Use the probability generating function of \(Z\) to find the most probable value of \(Z\).

5 The random variable \(X\) has the geometric distribution \(\operatorname { Geo } ( p )\).

1. Show that the probability generating function of \(X\) is \(\frac { \mathrm { pt } } { 1 - \mathrm { qt } }\), where \(\mathrm { q } = 1 - \mathrm { p }\).
2. Use the probability generating function of \(X\) to show that \(\operatorname { Var } ( X ) = \frac { \mathrm { q } } { \mathrm { p } ^ { 2 } }\).
   Kenny throws an ordinary fair 6-sided dice repeatedly. The random variable \(X\) is the number of throws that Kenny takes in order to obtain a 6 . The random variable \(Z\) denotes the sum of two independent values of \(X\).
3. Find the probability generating function of \(Z\).

3 Toby has a bag which contains 6 red marbles and 3 green marbles. He randomly chooses 3 marbles from the bag, without replacement. The random variable \(X\) is the number of red marbles that Toby obtains.

1. Find the probability generating function of \(X\).
   Ling also has a bag which contains 6 red marbles and 3 green marbles. He randomly chooses 2 marbles from his bag, without replacement. The random variable \(Y\) is the number of red marbles that Ling obtains. It is given that the probability generating function of \(Y\) is \(\frac { 1 } { 12 } \left( 1 + 6 t + 5 t ^ { 2 } \right)\). The random variable \(Z\) is the total number of red marbles that Toby and Ling obtain.
2. Find the probability generating function of \(Z\), expressing your answer as a polynomial in \(t\).
3. Use the probability generating function of \(Z\) to find \(\operatorname { Var } ( Z )\).