Expectation and variance of Poisson-related expressions

1 The number of goals scored by a team in a match is independent of other matches, and is denoted by the random variable \(X\), which has a Poisson distribution with mean 1.36. A supporter offers to make a donation of \(
) 5$ to the team for each goal that they score in the next 10 matches. Find the expectation and standard deviation of the amount that the supporter will pay.

3 In a golf tournament, the number of times in a day that a 'hole-in-one' is scored is denoted by the variable \(X\), which has a Poisson distribution with mean 0.15 . Mr Crump offers to pay \(
) 200$ each time that a hole-in-one is scored during 5 days of play. Find the expectation and variance of the amount that Mr Crump pays.

1 The random variable \(X\) has the distribution \(\operatorname { Po } ( 1.3 )\). The random variable \(Y\) is defined by \(Y = 2 X\).

1. Find the mean and variance of \(Y\).
2. Give a reason why the variable \(Y\) does not have a Poisson distribution.

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.

1 At a particular hospital, admissions of patients as a result of visits to the Accident and Emergency Department occur randomly at a uniform average rate of 0.75 per day. Independently, admissions that result from G.P. referrals occur randomly at a uniform average rate of 6.4 per week. The total number of admissions from these two causes over a randomly chosen period of four weeks is denoted by \(T\). State the distribution of \(T\) and obtain its expectation and variance.

6. Faults occur in a roll of material at a rate of \(\lambda\) per \(\mathrm { m } ^ { 2 }\). To estimate \(\lambda\), three pieces of material of sizes \(3 \mathrm {~m} ^ { 2 } , 7 \mathrm {~m} ^ { 2 }\) and \(10 \mathrm {~m} ^ { 2 }\) are selected and the number of faults \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) respectively are recorded. The estimator \(\hat { \lambda }\), where $$\hat { \lambda } = k \left( X _ { 1 } + X _ { 2 } + X _ { 3 } \right)$$ is an unbiased estimator of \(\lambda\).

1. Write down the distributions of \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) and find the value of \(k\).
2. Find \(\operatorname { Var } ( \hat { \lambda } )\). A random sample of \(n\) pieces of this material, each of size \(4 \mathrm {~m} ^ { 2 }\), was taken. The number of faults on each piece, \(Y\), was recorded.
3. Show that \(\frac { 1 } { 4 } \bar { Y }\) is an unbiased estimator of \(\lambda\).
4. Find \(\operatorname { Var } \left( \frac { 1 } { 4 } \bar { Y } \right)\).
5. Find the minimum value of \(n\) for which \(\frac { 1 } { 4 } \bar { Y }\) becomes a better estimator of \(\lambda\) than \(\hat { \lambda }\).

6. A spinner has edges numbered \(1,2,3,4\) and 5 . When the spinner is spun, the number of the edge on which it lands is the score. The probability distribution of the score, \(N\), is given in the table. It is known that \(\mathrm { E } ( N ) = 2.55\).

1. Find \(\operatorname { Var } ( N )\).
2. Find \(\mathrm { E } ( 3 N + 2 )\).
3. Find \(\operatorname { Var } ( 3 N + 2 )\).
   [0pt] [BLANK PAGE] A cloth manufacturer knows that faults occur randomly in the production process at a rate of 2 every 15 metres.
   (a) Find the probability of exactly 4 faults in a 15 metre length of cloth.
   (b) Find the probability of more than 10 faults in 60 metres of cloth. A retailer buys a large amount of this cloth and sells it in pieces of length \(x\) metres. He chooses \(x\) so that the probability of no faults in a piece is 0.80
   (c) Write down an equation for \(x\) and show that \(x = 1.7\) to 2 significant figures. The retailer sells 1200 of these pieces of cloth. He makes a profit of 60 p on each piece of cloth that does not contain a fault but a loss of \(\pounds 1.50\) on any pieces that do contain faults.
   (d) Find the retailer's expected profit.
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4. The numbers of customers arriving at a ticket desk between 8 a.m. and 9 a.m. on a Monday morning and on a Tuesday morning are denoted by \(X\) and \(Y\) respectively. It is given that \(X \sim \operatorname { Po } ( 17 )\) and \(Y \sim \operatorname { Po } ( 14 )\).

1. Find
   (a) \(\mathrm { P } ( X + Y ) > 40\),
   (b) \(\operatorname { Var } ( 2 X - Y )\).
2. State a necessary assumption for your calculations in part (i) to be valid.
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1 The numbers of customers arriving at a ticket desk between 8 a.m. and 9 a.m. on a Monday morning and on a Tuesday morning are denoted by \(X\) and \(Y\) respectively. It is given that \(X \sim \operatorname { Po } ( 17 )\) and \(Y \sim \operatorname { Po } ( 14 )\).

1. Find
   (a) \(\mathrm { P } ( X + Y ) > 40\),
   (b) \(\operatorname { Var } ( 2 X - Y )\).
2. State a necessary assumption for your calculations in part (i) to be valid.

1 The probability distribution for the discrete random variable \(W\) is given in the table.

1. Show that \(\operatorname { Var } ( W ) = 0.8571\).
2. Find \(\operatorname { Var } ( 3 W + 6 )\).