Poisson probabilities, mean and variance, and rescaling the rate Problem set A

\(X \sim \text{Po}(2.7)\). Use your calculator to find each probability.

1. (a) \(P(X = 0)\)
2. (b) \(P(X = 3)\)
3. (c) \(P(X = 5)\)

\(X \sim \text{Po}(6)\). Use your calculator to find each probability.

1. (a) \(P(X \le 4)\)
2. (b) \(P(X < 4)\)
3. (c) \(P(X \ge 7)\)
4. (d) \(P(X > 7)\)

\(X \sim \text{Po}(11)\). Write down the mean of X, then find the variance and the standard deviation of X.

1. (a) The mean, E(X).
2. (b) The variance, Var(X).
3. (c) The standard deviation.

Cracks appear in a road surface independently of each other, at a constant average rate of 2.2 per 100 metres. X is the number of cracks in a 100 metre stretch. Find the probability that the stretch has

1. (a) exactly 1 crack.
2. (b) at least 3 cracks.

A coastguard station receives distress calls independently of each other, at a constant average rate of 7.5 per week. X is the number of calls in a week. Write down the mean of X and find the standard deviation of X.

1. (a) The mean of X.
2. (b) The standard deviation of X.

A call centre receives calls independently of each other, at a constant average rate of 24 per hour. Find the probability that fewer than 6 calls arrive in a 15 minute period.

A bookshop sells copies of a novel independently of each other, at a constant average rate of 1.8 per day. Find the probability that at least 12 copies are sold in 5 days.

A production line has faults independently of each other, at a constant average rate of 5 per 4 hours. Find the probability that in a 1 hour period there are between 2 and 4 faults inclusive.