5.02a Discrete probability distributions: general

2 The probability generating function of the discrete random variable \(X\) is \(\frac { \mathrm { e } ^ { 4 t ^ { 2 } } } { \mathrm { e } ^ { 4 } }\). Find

1. \(\mathrm { E } ( X )\),
2. \(\mathrm { P } ( X = 2 )\). \(3 X _ { 1 }\) and \(X _ { 2 }\) are continuous random variables. Random samples of 5 observations of \(X _ { 1 }\) and 6 observations of \(X _ { 2 }\) are taken. No two observations are equal. The 11 observations are ranked, lowest first, and the sum of the ranks of the observations of \(X _ { 1 }\) is denoted by \(R\).

4 The moment generating function of a continuous random variable \(Y\), which has a \(\chi ^ { 2 }\) distribution with \(n\) degrees of freedom, is \(( 1 - 2 t ) ^ { - \frac { 1 } { 2 } n }\), where \(0 \leqslant t < \frac { 1 } { 2 }\).

1. Find \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\). For the case \(n = 1\), the sum of 60 independent observations of \(Y\) is denoted by \(S\).
2. Write down the moment generating function of \(S\) and hence identify the distribution of \(S\).
3. Use a normal approximation to estimate \(\mathrm { P } ( S \geqslant 70 )\).

4 The discrete random variable \(Y\) has probability generating function $$\mathrm { G } _ { Y } ( t ) = 0.09 t ^ { 2 } + 0.24 t ^ { 3 } + 0.34 t ^ { 4 } + 0.24 t ^ { 5 } + 0.09 t ^ { 6 }$$

1. Find the mean and variance of \(Y\). \(Y\) is the sum of two independent observations of a random variable \(X\).
2. Find the probability generating function of \(X\), expressing your answer as a cubic polynomial in \(t\).
3. Write down the value of \(\mathrm { P } ( X = 2 )\).

4 The random variable \(X\) has a \(\chi ^ { 2 }\) distribution with \(v\) degrees of freedom. The moment generating function of \(X\) is $$\mathrm { M } _ { X } ( t ) = ( 1 - 2 t ) ^ { - \frac { 1 } { 2 } v }$$

1. Show that \(\mathrm { E } ( X ) = v\).
2. Find \(\operatorname { Var } ( X )\).
3. Obtain the moment generating function of the sum \(Y\) of two independent \(\chi ^ { 2 }\) random variables, one with 6 degrees of freedom and the other with 8 degrees of freedom.
4. Identify the distribution of \(Y\).

6 In each round of a quiz a contestant can answer up to three questions. Each correct answer scores 1 point and allows the contestant to go on to the next question. A wrong answer scores 0 points and the contestant is allowed no further question in that round. If all 3 questions are answered correctly 1 bonus point is scored, making a total score of 4 for the round. For a certain contestant, \(A\), the probability of giving a correct answer is \(\frac { 3 } { 4 }\), independently of any other question. The random variable \(X _ { r }\) is the number of points scored by \(A\) during the \(r ^ { \text {th } }\) round.

1. Find the probability generating function of \(X _ { r }\).
2. Use the probability generating function found in part (i) to find the mean and variance of \(X _ { r }\).
3. Write down an expression for the probability generating function of \(X _ { 1 } + X _ { 2 }\) and find the probability that \(A\) has a total score of 4 at the end of two rounds.

2

1. The random variable \(Z\) has the standard Normal distribution with probability density function $$\mathrm { f } ( z ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - z ^ { 2 } / 2 } , \quad - \infty < z < \infty$$ Obtain the moment generating function of \(Z\).
2. Let \(\mathrm { M } _ { Y } ( t )\) denote the moment generating function of the random variable \(Y\). Show that the moment generating function of the random variable \(a Y + b\), where \(a\) and \(b\) are constants, is \(\mathrm { e } ^ { b t } \mathrm { M } _ { Y } ( a t )\).
3. Use the results in parts (i) and (ii) to obtain the moment generating function \(\mathrm { M } _ { X } ( t )\) of the random variable \(X\) having the Normal distribution with parameters \(\mu\) and \(\sigma ^ { 2 }\).
4. If \(W = \mathrm { e } ^ { X }\) where \(X\) is as in part (iii), \(W\) is said to have a lognormal distribution. Show that, for any positive integer \(k\), the expected value of \(W ^ { k }\) is \(\mathrm { M } _ { X } ( k )\). Use this result to find the expected value and variance of the lognormal distribution.

2 The random variable \(X\) has the \(\chi _ { n } ^ { 2 }\) distribution. This distribution has moment generating function \(\mathrm { M } ( \theta ) = ( 1 - 2 \theta ) ^ { - \frac { 1 } { 2 } n }\), where \(\theta < \frac { 1 } { 2 }\).

1. Verify the expression for \(\mathrm { M } ( \theta )\) quoted above for the cases \(n = 2\) and \(n = 4\), given that the probability density functions of \(X\) in these cases are as follows. $$\begin{array} { l l } n = 2 : & \mathrm { f } ( x ) = \frac { 1 } { 2 } \mathrm { e } ^ { - \frac { 1 } { 2 } x } \quad ( x > 0 ) \\ n = 4 : & \mathrm { f } ( x ) = \frac { 1 } { 4 } x \mathrm { e } ^ { - \frac { 1 } { 2 } x } \quad ( x > 0 ) \end{array}$$
2. For the general case, use \(\mathrm { M } ( \theta )\) to find the mean and variance of \(X\) in terms of \(n\).
3. \(Y _ { 1 } , Y _ { 2 } , \ldots , Y _ { k }\) are independent random variables, each with the \(\chi _ { 1 } ^ { 2 }\) distribution. Show that \(W = \sum _ { i = 1 } ^ { k } Y _ { i }\) has the \(\chi _ { k } ^ { 2 }\) distribution.
4. Use the Central Limit Theorem to find an approximation for \(\mathrm { P } ( W < 118.5 )\) for the case \(k = 100\).

A continuous random variable \(X\) is believed to have the probability density function f given by $$f ( x ) = \begin{cases} \frac { 3 } { 10 } \left( 5 x - x ^ { 2 } - 4 \right) & 2 \leqslant x < 4 \\ 0 & \text { otherwise } \end{cases}$$ A random sample of 60 observations was taken and these values are summarised in the following grouped frequency table. The estimated mean, based on the grouped data in the table above, is 2.69 , correct to 2 decimal places. It is decided that a goodness of fit test will only be conducted if the mean predicted from the probability density function is within \(10 \%\) of the estimated mean. Show that this condition is satisfied. The relevant expected frequencies are as follows. Show how the expected frequency for the interval \(3.2 \leqslant x < 3.6\) is obtained. Carry out the goodness of fit test at the 10\% significance level.

5 There is an insert for use in parts (iii) and (iv) of this question.
This question concerns the simulation of cars passing through two sets of pedestrian controlled traffic lights. The time intervals between cars arriving at the first set of lights are distributed according to Table 5.1. \begin{table}[h] \end{table}

1. Give an efficient rule for using two-digit random numbers to simulate arrival intervals.
2. Use two-digit random numbers from the list below to simulate the arrival times of five cars at the first lights. The first car arrives at the time given by the first arrival interval. Random numbers: \(24,01,99,89,77,19,58,42\) The two sets of traffic lights are 23 seconds driving time apart. Moving cars are always at least 2 seconds apart. If there is a queue at a set of lights, then when the red light ends the first car in the queue moves off immediately, the second car 2 seconds later, the third 2 seconds after that, etc. In this simple model there is to be no consideration of accelerations or decelerations, and the lights are either red or green. Table 5.2 shows the times when the lights are red. \begin{table}[h]

   \multirow{2}{*}{ first set of lights }	red start time	14	50	105	155
   \cline { 2 - 6 }	red end time	29	65	120	170
   \multirow{2}{*}{ second set of lights }	red start time	10	55	105	150
   \cline { 2 - 6 }	red end time	25	70	120	165

   \captionsetup{labelformat=empty} \caption{Table 5.2}
   \end{table}
3. Complete the table in the insert to simulate the passage of 10 cars through both sets of traffic lights. Use the arrival times given there.
4. Find the mean delay experienced by these cars in passing through each set of lights.
5. How could the output from this simulation model be made more reliable?

4 In a population colonizing an island 40\% of the first generation (parents) have brown eyes, \(40 \%\) have blue eyes and \(20 \%\) have green eyes. Offspring eye colour is determined according to the following rules. \section*{Eye colours of parents Eye colour of offspring} (1) both brown
(2) one brown and one blue \(50 \%\) brown and \(50 \%\) blue
(3) one brown and one green blue
(4) both blue \(25 \%\) brown, \(50 \%\) blue and \(25 \%\) green
(5) one blue and one green 50\% blue and \(50 \%\) green
(6) both green green

1. Give an efficient rule for using 1-digit random numbers to simulate the eye colour of a parent randomly selected from the colonizing population.
2. Give an efficient rule for using 1-digit random numbers to simulate the eye colour of offspring born of parents both of whom have blue eyes. The table in your answer book shows an incomplete simulation in which parent eye colours have been randomly selected, but in which offspring eye colours remain to be determined or simulated.
3. Complete the table using the given random numbers where needed. (You will need your own rules for cases \(( 2 )\) and 5 .)
   Each time you use a random number, explain how you decide which eye colour for the offspring. \(\square\)

4 A ski-lift gondola can carry 4 people. The weight restriction sign in the gondola says "4 people - 325 kg ". The table models the distribution of weights of people using the gondola.

1. Give an efficient rule for using 2-digit random numbers to simulate the weight of a person entering the gondola.
2. Give a reason for using 2-digit rather than 1-digit random numbers in these circumstances.
3. Using the random numbers given in your answer book, simulate the weights of four people entering the gondola, and hence give its simulated load.
4. Using the random numbers given in your answer book, repeat your simulation 9 further times. Hence estimate the probability of the load of a fully-laden gondola exceeding 325 kg .
5. What in reality might affect the pattern of loading of a gondola which is not modelled by your simulation?

12 The discrete random variable \(X\) takes values 1, 2, 3, 4 and 5, and its probability distribution is defined as follows. $$\mathrm { P } ( X = x ) = \begin{cases} a & x = 1 , \\ \frac { 1 } { 2 } \mathrm { P } ( X = x - 1 ) & x = 2,3,4,5 , \\ 0 & \text { otherwise } , \end{cases}$$ where \(a\) is a constant.

1. Show that \(a = \frac { 16 } { 31 }\). The discrete probability distribution for \(X\) is given in the table.

   \(x\)	1	2	3	4	5
   \(\mathrm { P } ( X = x )\)	\(\frac { 16 } { 31 }\)	\(\frac { 8 } { 31 }\)	\(\frac { 4 } { 31 }\)	\(\frac { 2 } { 31 }\)	\(\frac { 1 } { 31 }\)

2. Find the probability that \(X\) is odd. Two independent values of \(X\) are chosen, and their sum \(S\) is found.
3. Find the probability that \(S\) is odd.
4. Find the probability that \(S\) is greater than 8 , given that \(S\) is odd. Sheila sometimes needs several attempts to start her car in the morning. She models the number of attempts she needs by the discrete random variable \(Y\) defined as follows. $$\mathrm { P } ( Y = y + 1 ) = \frac { 1 } { 2 } \mathrm { P } ( Y = y ) \quad \text { for all positive integers } y .$$
5. Find \(\mathrm { P } ( Y = 1 )\).
6. Give a reason why one of the variables, \(X\) or \(Y\), might be more appropriate as a model for the number of attempts that Sheila needs to start her car.

12 A random variable \(X\) has probability distribution defined as follows. $$\mathrm { P } ( X = x ) = \begin{cases} k x & x = 1,2,3,4,5 , \\ 0 & \text { otherwise, } \end{cases}$$ where \(k\) is a constant.

1. Show that \(\mathrm { P } ( X = 3 ) = 0.2\).
2. Show in a table the values of \(X\) and their probabilities.
3. Two independent values of \(X\) are chosen, and their total \(T\) is found.
   1. Find \(\mathrm { P } ( T = 7 )\).
   2. Given that \(T = 7\), determine the probability that one of the values of \(X\) is 2 .

1. In a game, a player can score \(0,1,2,3\) or 4 points each time the game is played.

The random variable \(S\), representing the player's score, has the following probability distribution where \(a , b\) and \(c\) are constants. The probability of scoring less than 2 points is twice the probability of scoring at least 2 points. Each game played is independent of previous games played.
John plays the game twice and adds the two scores together to get a total.
Calculate the probability that the total is 6 points.

5. Manon has two biased spinners, one red and one green. The random variable \(R\) represents the score when the red spinner is spun.
The random variable \(G\) represents the score when the green spinner is spun.
The probability distributions for \(R\) and \(G\) are given below. Manon spins each spinner once and adds the two scores.

1. Find the probability that
   1. the sum of the two scores is 7
   2. the sum of the two scores is less than 4 The random variable \(X = m R + n G\) where \(m\) and \(n\) are integers. $$\mathrm { P } ( X = 20 ) = \frac { 1 } { 6 } \quad \text { and } \quad \mathrm { P } ( X = 50 ) = \frac { 1 } { 4 }$$
2. Find the value of \(m\) and the value of \(n\)

11 The discrete random variable \(X\) takes the values \(0,1,2,3,4\) and 5 with probabilities given by the formula $$\mathrm { P } ( X = x ) = k ( x + 1 ) ( 6 - x ) .$$

1. Find the value of \(k\). In one half-term Ben attends school on 40 days. The probability distribution above is used to model \(X\), the number of lessons per day in which Ben receives a gold star for excellent work.
2. Find the probability that Ben receives no gold stars on each of the first 3 days of the half-term and two gold stars on each of the next 2 days.
3. Find the expected number of days in the half-term on which Ben receives no gold stars.

1 Fig. 1 shows the probability distribution of the discrete random variable \(X\). \begin{table}[h] \end{table}

1. Find the value of \(k\).
2. Find \(\mathrm { P } ( X \neq 4 )\).

2 The discrete random variable \(X\) has probability distribution defined by $$\mathrm { P } ( X = x ) = \begin{cases} 0.1 & x = 0,1,2,3,4,5,6,7,8,9 \\ 0 & \text { otherwise } \end{cases}$$ Find the value of \(\mathrm { P } ( 4 \leq X \leq 7 )\) Circle your answer.

3 A discrete random variable \(X\) has the following probability distribution.

1. Determine the value of \(p\).
2. It is given that \(\mathrm { E } ( a X + b ) = \operatorname { Var } ( a X + b ) = 23.19\), where \(a\) and \(b\) are positive constants. Determine the value of \(a\) and the value of \(b\).

4 A discrete random variable \(W\) has the probability distribution shown in the following table, in which \(a\) and \(b\) are constants. It is given that \(\mathrm { E } ( W - 60 ) = 0.15\). Determine the value of \(\operatorname { Var } ( 4 W - 60 )\).

1 The random variable \(W\) can take values 1,2 or 3 and has a discrete uniform distribution.

1. Write down the value of \(\mathrm { E } ( 2 W )\).
2. Find the value of \(\operatorname { Var } ( 2 W )\).
3. Determine the value of the constant \(k\) for which \(\mathrm { E } ( 2 \mathrm {~W} + \mathrm { k } ) = \operatorname { Var } ( 2 \mathrm {~W} + \mathrm { k } )\). The random variable \(S\) has the probability distribution shown in the following table.

   \(S\)	2	3	4	5	6
   \(P ( S = S )\)	\(\frac { 2 } { 9 }\)	\(\frac { 1 } { 9 }\)	\(\frac { 1 } { 3 }\)	\(\frac { 1 } { 9 }\)	\(\frac { 2 } { 9 }\)

4. Calculate \(\operatorname { Var } ( S )\).

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

1. Find the value of \(\mathrm { E } ( A )\).
2. Determine the value of \(\operatorname { Var } ( A )\).
3. The variable \(A\) represents the value in pence of a coin chosen at random from a pile. Mia picks one coin at random from the pile. She then adds, from a different source, another coin of the same value as the one that she has chosen, and one 50p coin.
   1. Find the mean of the value of the three coins.
   2. Find the variance of the value of the three coins.

1 A discrete random variable \(X\) has the following distribution, where \(a , b\) and \(c\) are constants. It is given that \(\mathrm { E } ( X ) = 1.25\) and \(\operatorname { Var } ( X ) = 0.8875\).

1. Determine the values of \(a\), \(b\) and \(c\).
2. The random variable \(Y\) is defined by \(Y = 7 - 2 X\). Write down the value of \(\operatorname { Var } ( Y )\).
3. Twenty independent observations of \(X\) are obtained. The number of those observations for which \(X = 3\) is denoted by \(T\). Find the value of \(\operatorname { Var } ( T )\).

2 A discrete random variable \(D\) has the following probability distribution, where \(a\) is a constant. Determine the value of \(\operatorname { Var } ( 3 D + 4 )\).

3 A game is played as follows. A fair six-sided dice is thrown once. If the score obtained is even, the amount of money, in \(\pounds\), that the contestant wins is half the score on the dice, otherwise it is twice the score on the dice.

1. Find the probability distribution of the amount of money won by the contestant.
2. The contestant pays \(\pounds 5\) for every time the dice is thrown. Find the standard deviation of the loss made by the contestant in 120 throws of the dice.