Pure expectation and variance calculation

4 Wendy's journey to work consists of three parts: walking to the train station, riding on the train and then walking to the office. The times, in minutes, for the three parts of her journey are independent and have the distributions \(\mathrm { N } \left( 15.0,1.1 ^ { 2 } \right) , \mathrm { N } \left( 32.0,3.5 ^ { 2 } \right)\) and \(\mathrm { N } \left( 8.6,1.2 ^ { 2 } \right)\) respectively.

1. Find the mean and variance of the total time for Wendy's journey.
   If Wendy's journey takes more than 60 minutes, she is late for work.
2. Find the probability that, on a randomly chosen day, Wendy will be late for work.
3. Find the probability that the mean of Wendy's journey times over 15 randomly chosen days will be less than 54.5 minutes.

4 The independent random variables \(X\) and \(Y\) have distributions \(\operatorname { Po } ( 2 )\) and \(\mathrm { B } \left( 20 , \frac { 1 } { 4 } \right)\) respectively.

1. Find the mean and standard deviation of \(X - 3 Y\).
2. Find \(\mathrm { P } ( Y = 15 X )\).

3 The independent random variables \(X\) and \(Y\) are such that \(X\) has mean 8 and variance 4.8 and \(Y\) has a Poisson distribution with mean 6. Find

1. \(\mathrm { E } ( 2 X - 3 Y )\),
2. \(\operatorname { Var } ( 2 X - 3 Y )\).

7 The independent variables \(X\) and \(Y\) are such that \(X \sim \mathrm {~B} ( 10,0.8 )\) and \(Y \sim \mathrm { Po } ( 3 )\). Find

1. \(\mathrm { E } ( 7 X + 5 Y - 2 )\),
2. \(\operatorname { Var } ( 4 X - 3 Y + 3 )\),
3. \(\mathrm { P } ( 2 X - Y = 18 )\).

1 The independent random variables \(X\) and \(Y\) have standard deviations 3 and 6 respectively. Calculate the standard deviation of \(4 X - 5 Y\).

1 A fair six-sided die is thrown 20 times and the number of sixes, \(X\), is recorded. Another fair six-sided die is thrown 20 times and the number of odd-numbered scores, \(Y\), is recorded. Find the mean and standard deviation of \(X + Y\).

1 The random variables \(X\) and \(Y\) are independent with \(X \sim \operatorname { Po } ( 5 )\) and \(Y \sim \operatorname { Po } ( 4 )\). \(S\) denotes the sum of 2 observations of \(X\) and 3 observations of \(Y\).

1. Find \(\mathrm { E } ( S )\) and \(\operatorname { Var } ( S )\).
2. The random variable \(T\) is defined by \(\frac { 1 } { 2 } X - \frac { 1 } { 4 } Y\). Show that \(\mathrm { E } ( T ) = \operatorname { Var } ( T )\).
3. State which of \(S\) and \(T\) (if either) does not have a Poisson distribution, giving a reason for your answer.

1 The independent random variables \(X\) and \(Y\) have the distributions \(\mathrm { N } \left( 10 , \sigma ^ { 2 } \right)\) and \(\operatorname { Po } ( 2 )\) respectively. The random variable \(S\) is given by \(S = 5 X - 2 Y + c\), where \(c\) is a constant.
It is given that \(\mathrm { E } ( S ) = \operatorname { Var } ( S ) = 408\).

1. Find the value of \(c\) and show that \(\sigma ^ { 2 } = 16\).
2. Find \(\mathrm { P } ( X \geqslant \mathrm { E } ( Y ) )\).

5 The discrete random variables \(X\) and \(Y\) are independent with \(X \sim \mathrm {~B} \left( 32 , \frac { 1 } { 2 } \right)\) and \(Y \sim \operatorname { Po } ( 28 )\).

1. Find the values of \(\mathrm { E } ( Y - X )\) and \(\operatorname { Var } ( Y - X )\).
2. State, with justification, an approximate distribution for \(Y - X\).
3. Hence find \(\mathrm { P } ( | Y - X | \geqslant 3 )\).

2 Natasha is investigating binomial distributions. She constructs the spreadsheet in Fig. 2 which shows the first 3 and last 4 rows of a simulation involving two independent variables, \(X\) and \(Y\), with distributions \(\mathrm { B } ( 10,0.3 )\) and \(\mathrm { B } ( 50,0.3 )\) respectively. The spreadsheet also shows the corresponding value of the random variable \(Z\), defined by \(Z = 5 X - Y\), for each pair of values of \(X\) and \(Y\). There are 100 simulated values of each of \(X , Y\) and \(Z\). The spreadsheet also shows whether each value of \(Z\) is greater than 6, and cells D103 and D104 show the number of values of \(Z\) which are greater than 6 and not greater than 6 respectively. \begin{table}[h] \end{table}

1. Use the information in the spreadsheet to write down an estimate of \(\mathrm { P } ( Z > 6 )\).
2. Explain how a more reliable estimate of \(\mathrm { P } ( Z > 6 )\) could be obtained.
3. 1. State the greatest possible value of \(Z\).
   2. Explain why it is very unlikely that \(Z\) would have this value.
4. Use the Central Limit Theorem to calculate an estimate of the probability that the mean of 20 independent values of \(Z\) is greater than 2 .

2 The independent random variables \(X\) and \(Y\) have normal distributions where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 3 \mu , 4 \sigma ^ { 2 } \right)\). Two random samples each of size \(n\) are taken, one from each of these normal populations.

1. Show that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\) provided that \(a + 3 b = 1\), where \(a\) and \(b\) are constants and \(\bar { X }\) and \(\bar { Y }\) are the respective sample means. In the remainder of the question assume that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\).
2. Show that \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) can be written as \(\frac { \sigma ^ { 2 } } { n } \left( 1 - 6 b + 13 b ^ { 2 } \right)\).
3. The value of the constant \(b\) can be varied. Find the value of \(b\) that gives the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\), and hence find the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) in terms of \(\sigma\) and \(n\).

4 The independent random variables \(X\) and \(Y\) have normal distributions where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 3 \mu , 4 \sigma ^ { 2 } \right)\). Two random samples each of size \(n\) are taken, one from each of these normal populations.

1. Show that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\) provided that \(a + 3 b = 1\), where \(a\) and \(b\) are constants and \(\bar { X }\) and \(\bar { Y }\) are the respective sample means. In the remainder of the question assume that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\).
2. Show that \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) can be written as \(\frac { \sigma ^ { 2 } } { n } \left( 1 - 6 b + 13 b ^ { 2 } \right)\).
3. The value of the constant \(b\) can be varied. Find the value of \(b\) that gives the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\), and hence find the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) in terms of \(\sigma\) and \(n\).

4 The independent random variables \(X\) and \(Y\) have normal distributions where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 3 \mu , 4 \sigma ^ { 2 } \right)\). Two random samples each of size \(n\) are taken, one from each of these normal populations.

1. Show that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\) provided that \(a + 3 b = 1\), where \(a\) and \(b\) are constants and \(\bar { X }\) and \(\bar { Y }\) are the respective sample means. In the remainder of the question assume that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\).
2. Show that \(\operatorname { Var } ( \overline { a X } + b \bar { Y } )\) can be written as \(\frac { \sigma ^ { 2 } } { n } \left( 1 - 6 b + 13 b ^ { 2 } \right)\).
3. The value of the constant \(b\) can be varied. Find the value of \(b\) that gives the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\), and hence find the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) in terms of \(\sigma\) and \(n\).

4 The independent random variables \(X\) and \(Y\) have normal distributions where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 3 \mu , 4 \sigma ^ { 2 } \right)\). Two random samples each of size \(n\) are taken, one from each of these normal populations.

1. Show that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\) provided that \(a + 3 b = 1\), where \(a\) and \(b\) are constants and \(\bar { X }\) and \(\bar { Y }\) are the respective sample means. In the remainder of the question assume that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\).
2. Show that \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) can be written as \(\frac { \sigma ^ { 2 } } { n } \left( 1 - 6 b + 13 b ^ { 2 } \right)\).
3. The value of the constant \(b\) can be varied. Find the value of \(b\) that gives the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\), and hence find the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) in terms of \(\sigma\) and \(n\).

The weights of bags of fuel have mean 3.2 kg and standard deviation 0.04 kg. The total weight of a random sample of three bags is denoted by \(T\) kg. Find the mean and standard deviation of \(T\). [4]

Ben is a fencing contractor who is often required to repair a garden fence by replacing a broken post between fence panels, as illustrated. \includegraphics{figure_4} The tasks involved are as follows. \(U\): detach the two fence panels from the broken post \(V\): remove the broken post \(W\): insert a new post \(X\): attach the two fence panels to the new post The mean and the standard deviation of the time, in minutes, for each of these tasks are shown in the table.

Task	Mean	Standard deviation
\(U\)	15	5
\(V\)	40	15
\(W\)	75	20
\(X\)	20	10

The random variables \(U\), \(V\), \(W\) and \(X\) are pairwise independent, except for \(V\) and \(W\) for which \(\rho_{VW} = 0.25\).

1. Determine values for the mean and the variance of:
   1. \(R = U + X\);
   2. \(F = V + W\);
   3. \(T = R + F\);
   4. \(D = W - V\).
   [8 marks]
2. Assuming that each of \(R\), \(F\), \(T\) and \(D\) is approximately normally distributed, determine the probability that:
   1. the total time taken by Ben to repair a garden fence is less than 3 hours;
   2. the time taken by Ben to insert a new post is at least 1 hour more than the time taken by him to remove the broken post.
   [5 marks]