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

\(X \sim \text{Po}(3.4)\). Use your calculator to find each probability.

1. (a) \(P(X = 1)\)
2. (b) \(P(X = 2)\)
3. (c) \(P(X = 6)\)

\(X \sim \text{Po}(7.5)\). Use your calculator to find each probability.

1. (a) \(P(X \le 5)\)
2. (b) \(P(X < 5)\)
3. (c) \(P(X \ge 9)\)
4. (d) \(P(X > 9)\)

\(X \sim \text{Po}(13)\). 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.

Flaws occur in a roll of fabric independently of each other, at a constant average rate of 3.1 per 10 metre length. X is the number of flaws in a 10 metre length. Find the probability that the length has

1. (a) exactly 4 flaws.
2. (b) at least 2 flaws.

A weather station records hailstorms independently of each other, at a constant average rate of 5.5 per month. X is the number of hailstorms in a month. 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 ferry crew sees dolphins independently of each other, at a constant average rate of 18 per hour. Find the probability that fewer than 4 dolphins are seen in a 10 minute period.

A nesting box is visited by birds independently of each other, at a constant average rate of 0.7 per hour. Find the probability that more than 5 visits are made in 6 hours.

A trail camera photographs deer independently of each other, at a constant average rate of 9 per 6 hours. Find the probability that in a 2 hour period there are between 2 and 5 photographs inclusive.