5.04a Linear combinations: E(aX+bY), Var(aX+bY)

3 From the records of Mulcaster United Football Club the following distribution was suggested as a probability model for future matches. \(X\) and \(Y\) denoted the numbers of goals scored by the home team and the away team respectively. Use the model to find

1. \(\mathrm { E } ( X )\),
2. the probability that the away team wins a randomly chosen match,
3. the probability that the away team wins a randomly chosen match, given that the home team scores. One of the directors, an amateur statistician, finds that \(\operatorname { Cov } ( X , Y ) = 0.007\). He states that, as this value is very close to zero, \(X\) and \(Y\) may be considered to be independent.
4. Comment on the director's statement.

6 A City Council comprises 16 Labour members, 14 Conservative members and 6 members of Other parties. A sample of two members was chosen at random to represent the Council at an event. The number of Labour members and the number of Conservative members in this sample are denoted by \(L\) and \(C\) respectively. The joint probability distribution of \(L\) and \(C\) is given in the following table. \(C\)

\(L\)
	0	1	2
0	\(\frac { 1 } { 42 }\)	\(\frac { 16 } { 105 }\)	\(\frac { 4 } { 21 }\)
1	\(\frac { 2 } { 15 }\)	\(\frac { 16 } { 45 }\)	0
2	\(\frac { 13 } { 90 }\)	0	0

1. Verify the two non-zero probabilities in the table for which \(C = 1\).
2. Find the expected number of Conservatives in the sample.
3. Find the expected number of Other members in the sample.
4. Explain why \(L\) and \(C\) are not independent, and state what can be deduced about \(\operatorname { Cov } ( L , C )\).

6 The random variables \(S\) and \(T\) are independent and have joint probability distribution given in the table.

1. Show that \(a = 0.12\) and find the value of \(b\).
2. Find \(\mathrm { P } ( T - S = 1 )\).
3. Find \(\operatorname { Var } ( T - S )\).

1 An unbiased coin is tossed three times. The random variables \(F\) and \(S\) denote the total number of heads that occur in the first two tosses and the total number of heads that occur in the last two tosses respectively. The table above shows the joint probability distribution of \(F\) and \(S\).

1. Show how the entry \(\frac { 1 } { 4 }\) in the table is obtained.
2. Find \(\operatorname { Cov } ( F , S )\).

5 Two discrete random variables \(X\) and \(Y\) have a joint probability distribution defined by $$\mathrm { P } ( X = x , Y = y ) = a ( x + y + 1 ) \quad \text { for } x = 0,1,2 \text { and } y = 0,1,2 ,$$ where \(a\) is a constant.

1. Show that \(a = \frac { 1 } { 27 }\).
2. Find \(\mathrm { E } ( X )\).
3. Find \(\operatorname { Cov } ( X , Y )\).
4. Are \(X\) and \(Y\) independent? Give a reason for your answer.
5. Find \(\mathrm { P } ( X = 1 \mid Y = 2 )\).

2 [In this question, you may use the result \(\int _ { 0 } ^ { \infty } u ^ { m } \mathrm { e } ^ { - u } \mathrm {~d} u = m\) ! for any non-negative integer \(m\).]
The random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} \frac { \lambda ^ { k + 1 } x ^ { k } \mathrm { e } ^ { - \lambda x } } { k ! } , & x > 0 \\ 0 , & \text { elsewhere } \end{cases}$$ where \(\lambda > 0\) and \(k\) is a non-negative integer.

1. Show that the moment generating function of \(X\) is \(\left( \frac { \lambda } { \lambda - \theta } \right) ^ { k + 1 }\).
2. The random variable \(Y\) is the sum of \(n\) independent random variables each distributed as \(X\). Find the moment generating function of \(Y\) and hence obtain the mean and variance of \(Y\). [8]
3. State the probability density function of \(Y\).
4. For the case \(\lambda = 1 , k = 2\) and \(n = 5\), it may be shown that the definite integral of the probability density function of \(Y\) between limits 10 and \(\infty\) is 0.9165 . Calculate the corresponding probability that would be given by a Normal approximation and comment briefly.

2 Independent trials, on each of which the probability of a 'success' is \(p ( 0 < p < 1 )\), are being carried out. The random variable \(X\) counts the number of trials up to and including that on which the first success is obtained. The random variable \(Y\) counts the number of trials up to and including that on which the \(n\)th success is obtained.

1. Write down an expression for \(\mathrm { P } ( X = x )\) for \(x = 1,2 , \ldots\). Show that the probability generating function of \(X\) is $$\mathrm { G } ( t ) = p t ( 1 - q t ) ^ { - 1 }$$ where \(q = 1 - p\), and hence that the mean and variance of \(X\) are $$\mu = \frac { 1 } { p } \quad \text { and } \quad \sigma ^ { 2 } = \frac { q } { p ^ { 2 } }$$ respectively.
2. Explain why the random variable \(Y\) can be written as $$Y = X _ { 1 } + X _ { 2 } + \ldots + X _ { n }$$ where the \(X _ { i }\) are independent random variables each distributed as \(X\). Hence write down the probability generating function, the mean and the variance of \(Y\).
3. State an approximation to the distribution of \(Y\) for large \(n\).
4. The aeroplane used on a certain flight seats 140 passengers. The airline seeks to fill the plane, but its experience is that not all the passengers who buy tickets will turn up for the flight. It uses the random variable \(Y\) to model the situation, with \(p = 0.8\) as the probability that a passenger turns up. Find the probability that it needs to sell at least 160 tickets to get 140 passengers who turn up. Suggest a reason why the model might not be appropriate.

1 In a certain country, any baby born is equally likely to be a boy or a girl, independently for all births. The birthweight of a baby boy is given by the continuous random variable \(X _ { B }\) with probability density function (pdf) \(\mathrm { f } _ { B } ( x )\) and cumulative distribution function (cdf) \(\mathrm { F } _ { B } ( x )\). The birthweight of a baby girl is given by the continuous random variable \(X _ { G }\) with pdf \(\mathrm { f } _ { G } ( x )\) and cdf \(\mathrm { F } _ { G } ( x )\). The continuous random variable \(X\) denotes the birthweight of a baby selected at random.

1. By considering $$\mathrm { P } ( X \leqslant x ) = \mathrm { P } ( X \leqslant x \mid \text { boy } ) \mathrm { P } ( \text { boy } ) + \mathrm { P } ( X \leqslant x \mid \text { girl } ) \mathrm { P } ( \text { girl } ) ,$$ find the cdf of \(X\) in terms of \(\mathrm { F } _ { B } ( x )\) and \(\mathrm { F } _ { G } ( x )\), and deduce that the pdf of \(X\) is $$\mathrm { f } ( x ) = \frac { 1 } { 2 } \left\{ \mathrm { f } _ { B } ( x ) + \mathrm { f } _ { G } ( x ) \right\} .$$
2. The birthweights of baby boys and girls have means \(\mu _ { B }\) and \(\mu _ { G }\) respectively. Deduce that $$\mathrm { E } ( X ) = \frac { 1 } { 2 } \left( \mu _ { B } + \mu _ { G } \right) .$$
3. The birthweights of baby boys and girls have common variance \(\sigma ^ { 2 }\). Find an expression for \(\mathrm { E } \left( X ^ { 2 } \right)\) in terms of \(\mu _ { B } , \mu _ { G }\) and \(\sigma ^ { 2 }\), and deduce that $$\operatorname { Var } ( X ) = \sigma ^ { 2 } + \frac { 1 } { 4 } \left( \mu _ { B } - \mu _ { G } \right) ^ { 2 } .$$
4. A random sample of size \(2 n\) is taken from all the babies born in a certain period. The mean birthweight of the babies in this sample is \(\bar { X }\). Write down an approximation to the sampling distribution of \(\bar { X }\) if \(n\) is large.
5. Suppose instead that a stratified sample of size \(2 n\) is taken by selecting \(n\) baby boys at random and, independently, \(n\) baby girls at random. The mean birthweight of the \(2 n\) babies in this sample is \(\bar { X } _ { s t }\). Write down the expected value of \(\bar { X } _ { s t }\) and find the variance of \(\bar { X } _ { s t }\).
6. Deduce that both \(\bar { X }\) and \(\bar { X } _ { s t }\) are unbiased estimators of the population mean birthweight. Find which is the more efficient.

2

1. The probability density function of the random variable \(X\) is $$\mathrm { f } ( x ) = \frac { x ^ { k - 1 } \mathrm { e } ^ { - x / \phi } } { \phi ^ { k } ( k - 1 ) ! } , x > 0$$ where \(k\) is a known positive integer and \(\phi\) is an unknown parameter ( \(\phi > 0\) ). Show that the moment generating function (mgf) of \(X\) is $$\mathrm { M } _ { X } ( \theta ) = ( 1 - \phi \theta ) ^ { - k }$$ for \(\theta < \frac { 1 } { \phi }\).
2. Write down the mgf of the random variable \(W = \sum _ { i = 1 } ^ { n } X _ { i }\) where \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) are independent random variables each with the same distribution as \(X\).
3. Write down the mgf of the random variable \(Y = \frac { 2 W } { \phi }\). Given that the mgf of the random variable \(V\) having the \(\chi _ { m } ^ { 2 }\) distribution is \(\mathrm { M } _ { V } ( \theta ) = ( 1 - 2 \theta ) ^ { - m / 2 }\) (for \(\theta < \frac { 1 } { 2 }\) ), deduce the distribution of \(Y\).
4. Deduce that \(\mathrm { P } \left( l < \frac { 2 W } { \phi } < u \right) = 0.95\) where \(l\) and \(u\) are the lower and upper \(2 \frac { 1 } { 2 } \%\) points of the \(\chi _ { 2 n k } ^ { 2 }\) distribution. Hence deduce that a \(95 \%\) confidence interval for \(\phi\) is given by \(\left( \frac { 2 w } { u } , \frac { 2 w } { l } \right)\) where \(w\) is an observation on the random variable \(W\).
5. For the case \(k = 2\) and \(n = 10\), use percentage points of the \(\chi ^ { 2 }\) distribution to write down, in terms of \(w\), an expression for a \(95 \%\) confidence interval for \(\phi\). By considering the \(\operatorname { mgf }\) of \(W\), find in terms of \(\phi\) the expected length of this interval.

2 The random variable \(Z\) has the standard Normal distribution. The random variable \(Y\) is defined by \(Y = Z ^ { 2 }\).
You are given that \(Y\) has the following probability density function. $$\mathrm { f } ( y ) = \frac { 1 } { \sqrt { 2 \pi y } } \mathrm { e } ^ { - \frac { 1 } { 2 } y } , \quad y > 0$$

1. Show that the moment generating function (mgf) of \(Y\) is given by $$\mathrm { M } _ { Y } ( \theta ) = ( 1 - 2 \theta ) ^ { - \frac { 1 } { 2 } }$$
2. Use the mgf to obtain \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\). The random variable \(U\) is defined by $$U = Z _ { 1 } ^ { 2 } + Z _ { 2 } ^ { 2 } + \ldots + Z _ { n } ^ { 2 } ,$$ where \(Z _ { 1 } , Z _ { 2 } , \ldots , Z _ { n }\) are independent standard Normal random variables.
3. State an appropriate general theorem for mgfs and hence write down the mgf of \(U\). State the values of \(\mathrm { E } ( U )\) and \(\operatorname { Var } ( U )\). The random variable \(W\) is defined by $$W = \frac { U - n } { \sqrt { 2 n } }$$
4. Show that the logarithm of the \(\operatorname { mgf }\) of \(W\) is $$- \sqrt { \frac { n } { 2 } } \theta - \frac { n } { 2 } \ln \left( 1 - \sqrt { \frac { 2 } { n } } \theta \right) .$$ Use the series expansion of \(\ln ( 1 - t )\) to show that, as \(n \rightarrow \infty\), this expression tends to \(\frac { 1 } { 2 } \theta ^ { 2 }\).
   State what this implies about the distribution of \(W\) for large \(n\).

2 The random variable \(X\) has probability density function \(\mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \lambda \mathrm { e } ^ { - \lambda x } , \quad x > 0 .$$

1. Obtain the moment generating function (mgf) of \(X\).
2. Use the mgf to find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\). The random variable \(Y\) is defined as follows: $$Y = X _ { 1 } + \ldots + X _ { n } ,$$ where the \(X _ { i }\) are independently and identically distributed as \(X\).
3. Write down expressions for \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\). Obtain the \(\operatorname { mgf }\) of \(Y\).
4. Find the \(\operatorname { mgf }\) of \(Z\) where \(Z = \frac { Y - \frac { n } { \lambda } } { \frac { \sqrt { n } } { \lambda } }\).
5. By considering the logarithm of the mgf of \(Z\), show that the distribution of \(Z\) tends to the standard Normal distribution as \(n\) tends to infinity.

5. The discrete random variable \(X\) has the following probability distribution where \(a\), \(b\) and \(c\) are probabilities.
Given that \(\mathrm { E } ( X ) = 0.8\)

1. find the value of \(c\). Given also that \(\mathrm { E } \left( X ^ { 2 } \right) = 5\) find
2. the value of \(a\) and the value of \(b\),
3. \(\operatorname { Var } ( X )\) The random variable \(Y = 5 - 3 X\) Find
4. \(\mathrm { E } ( Y )\)
5. \(\operatorname { Var } ( Y )\)
6. \(\mathrm { P } ( Y \geqslant 0 )\)

7 A large railway network suffers points failures at an average rate of 1 every 3 days. Assume that the number of points failures can be modelled by a Poisson distribution. The network employs a new firm of engineers. After the new engineers have become established, it is found that in a randomly chosen period of 15 days there are 2 instances of points failures.

1. Test, at the \(5 \%\) significance level, whether there is evidence that the mean number of points failures has been reduced.
2. A new test is carried out over a period of 150 days. Use a suitable approximation to find the greatest number of points failures there could be in 150 days that would lead to a \(5 \%\) significance test concluding that the average number of points failures had been reduced.

1 The independent random variables \(X\) and \(Y\) have Poisson distributions with parameters 16 and 2 respectively, and \(Z = \frac { 1 } { 2 } X - Y\).

1. Find \(\mathrm { E } ( Z )\) and \(\operatorname { Var } ( Z )\).
2. State whether \(Z\) has a Poisson distribution, giving a reason for your answer.

5 Two guesthouses, the Albion and the Blighty, have 8 and 6 rooms respectively. The demand for rooms at the Albion has a Poisson distribution with mean 6.5 and the demand for rooms at the Blighty has an independent Poisson distribution with mean 5.5. The owners have agreed that if their guesthouse is full, they will re-direct guests to the other.

1. Find the probability that, on any particular night, the two guesthouses together do not have enough rooms to meet demand.
2. The Albion charges \(\pounds 60\) per room per night, and the Blighty \(\pounds 80\). Find the probability, that on a particular night, the total income of the two guesthouses is exactly \(\pounds 400\).
3. If \(A\) is the number of rooms demanded at the Albion each night, and \(B\) the number of rooms demanded at the Blighty each night, find the mean and variance of the variable \(C = 60 A + 80 B\). State whether \(C\) has a Poisson distribution, giving a reason for your answer.

3 The table shows the joint probability distribution of two random variables \(X\) and \(Y\).

1. Find \(\operatorname { Cov } ( X , Y )\).
2. Are \(X\) and \(Y\) independent? Give a reason for your answer.
3. Find \(\mathrm { P } ( X = 1 \mid X Y = 2 )\).

6 Andrew has five coins. Three of them are unbiased. The other two are biased such that the probability of obtaining a head when one of them is tossed is \(\frac { 3 } { 5 }\). Andrew tosses all five coins. It is given that the probability generating function of \(X\), the number of heads obtained on the unbiased coins, is \(\mathrm { G } _ { X } ( t )\), where $$\mathrm { G } _ { X } ( t ) = \frac { 1 } { 8 } + \frac { 3 } { 8 } t + \frac { 3 } { 8 } t ^ { 2 } + \frac { 1 } { 8 } t ^ { 3 }$$

1. Find \(G _ { Y } ( \mathrm { t } )\), the probability generating function of \(Y\), the number of heads on the biased coins.
2. The random variable \(Z\) is the total number of heads obtained when Andrew tosses all five coins. Find the probability generating function of \(Z\), giving your answer as a polynomial.
3. Find \(\mathrm { E } ( Z )\) and \(\operatorname { Var } ( Z )\).
4. Write down the value of \(\mathrm { P } ( Z = 3 )\).

2 The independent discrete random variables \(X\) and \(Y\) can take the values 0,1 and 2 with probabilities as given in the tables. \(\quad\)

\(y\)	0	1	2
\(\mathrm { P } ( Y = y )\)	0.5	0.3	0.2

The random variables \(U\) and \(V\) are defined as follows: $$U = X Y , V = | X - Y | .$$

1. In the Printed Answer Book complete the table giving the joint distribution of \(U\) and \(V\).
2. Find \(\operatorname { Cov } ( U , V )\).
3. Find \(\mathrm { P } ( U V = 0 \mid V = 2 )\).

5 The discrete random variable \(X\) is such that \(\mathrm { P } ( X = x ) = \frac { 3 } { 4 } \left( \frac { 1 } { 4 } \right) ^ { x } , x = 0,1,2 , \ldots\).

1. Show that the moment generating function of \(X , \mathrm { M } _ { X } ( t )\), can be written as \(\mathrm { M } _ { X } ( t ) = \frac { 3 } { 4 - \mathrm { e } ^ { t } }\).
2. Find the range of values of \(t\) for which the formula for \(\mathrm { M } _ { X } ( t )\) in part (i) is valid.
3. Use \(\mathrm { M } _ { X } ( t )\) to find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

4 Each time Ben attempts to complete a crossword in his daily newspaper, the probability that he succeeds is \(\frac { 2 } { 3 }\). The random variable \(X\) denotes the number of times that Ben succeeds in 9 attempts.

1. Find
   1. \(\mathrm { P } ( X = 6 )\),
   2. \(\mathrm { P } ( X < 6 )\),
   3. \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\). Ben notes three values, \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\), of \(X\).
   4. State the total number of attempts to complete a crossword that are needed to obtain three values of \(X\). Hence find \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } + X _ { 3 } = 18 \right)\).

3 Tennis balls are dropped from a standard height, and the height of bounce, \(H \mathrm {~cm}\), is measured. \(H\) is a random variable with the distribution \(\mathrm { N } \left( 40 , \sigma ^ { 2 } \right)\). It is given that \(\mathrm { P } ( H < 32 ) = 0.2\).

1. Find the value of \(\sigma\).
2. 90 tennis balls are selected at random. Use an appropriate approximation to find the probability that more than 19 have \(H < 32\).

6

1. The random variable \(D\) has the distribution \(\operatorname { Po } ( 24 )\). Use a suitable approximation to find \(P ( D > 30 )\).
2. An experiment consists of 200 trials. For each trial, the probability that the result is a success is 0.98 , independent of all other trials. The total number of successes is denoted by \(E\).
   1. Explain why the distribution of \(E\) cannot be well approximated by a Poisson distribution.
   2. By considering the number of failures, use an appropriate Poisson approximation to find \(\mathrm { P } ( E \leqslant 194 )\).

7 The number of customer complaints received by a company per day is denoted by \(X\). Assume that \(X\) has the distribution \(\operatorname { Po } ( 2.2 )\).

1. In a week of 5 working days, the probability there are at least \(n\) customer complaints is 0.146 correct to 3 significant figures. Use tables to find the value of \(n\).
2. Use a suitable approximation to find the probability that in a period of 20 working days there are fewer than 38 customer complaints. A week of 5 working days in which at least \(n\) customer complaints are received, where \(n\) is the value found in part (i), is called a 'bad' week.
3. Use a suitable approximation to find the probability that, in 40 randomly chosen weeks, more than 7 are bad.

2

1. For the continuous random variable \(V\), it is known that \(\mathrm { E } ( V ) = 72.0\). The mean of a random sample of 40 observations of \(V\) is denoted by \(\bar { V }\). Given that \(\mathrm { P } ( \bar { V } < 71.2 ) = 0.35\), estimate the value of \(\operatorname { Var } ( V )\).
2. Explain why you need to use the Central Limit Theorem in part (i), and why its use is justified.