The sum of two independent Poisson variables Problem set B

X and Y are independent, with \(X \sim \text{Po}(2.8)\) and \(Y \sim \text{Po}(4.6)\). Let Z = X + Y.

1. (a) Write down the distribution of Z.
2. (b) Find \(P(Z = 6)\).
3. (c) Find \(P(Z > 8)\).

For each situation, give one reason why the assumptions of a constant average rate and independence may not hold.

1. (a) The number of calls a pizza restaurant receives each hour for delivery, and the number it receives for collection, modelled as independent Poisson variables.
2. (b) The number of pupils in a school who catch a stomach bug each week, and the number who catch a cold, modelled as independent Poisson variables.

Parcels arrive at a sorting office from depot P at a constant average rate of 6.3 per hour and from depot Q at 2.9 per hour. Assume the two counts are independent Poisson variables. Find the probability that more than 9 parcels arrive in one hour.

A bike shop sells helmets at a constant average rate of 2 per hour and lights at a constant average rate of 9 per 3 hours. Assume the two counts are independent Poisson variables. Find the probability that at least 16 helmets and lights are sold in 3 hours.

A library issues fiction books at a constant average rate of 15 per hour and non-fiction books at a constant average rate of 11 per hour. Assume the two counts are independent Poisson variables. Z is the total number of books per hour.

1. (a) Write down the distribution of Z.
2. (b) Find \(P(Z \le 22)\).
3. (c) Give two reasons why the assumption that the counts are independent Poisson variables is unlikely to hold.

Two independent groups of visitors reach a museum entrance: group R at a constant average rate of 3.9 per minute and group S at 5.6 per minute. Assume both counts are Poisson variables. Z is the total number per minute. Find the mean and standard deviation of Z, and the probability that Z is within one standard deviation of its mean.

1. (a) The mean of Z.
2. (b) The standard deviation of Z.
3. (c) The probability that Z is within one standard deviation of its mean.

X and Y are independent Poisson variables. Y has mean 8, and Z = X + Y has mean 12.5. Find the mean of X, and find P(Z ≥ 16).

1. (a) The mean of X.
2. (b) \(P(Z \ge 16)\)