Poisson probabilities, mean and variance, and rescaling the rate Problem set A One question at a time

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

1. (a) \(P(X = 0)\)

   • From the calculator: Poisson pdf with λ = 2.7, x = 0
   • P(X = 0) = 0.0672

2. (b) \(P(X = 3)\)

   • From the calculator: Poisson pdf with λ = 2.7, x = 3
   • P(X = 3) = 0.220

3. (c) \(P(X = 5)\)

   • From the calculator: Poisson pdf with λ = 2.7, x = 5
   • P(X = 5) = 0.0804

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

1. (a) \(P(X \le 4)\)

   • From the calculator: Poisson cdf with λ = 6, lower = 0, upper = 4
   • P(X ≤ 4) = 0.285

2. (b) \(P(X < 4)\)

   • Fewer than 4 means at most 3
   • From the calculator: Poisson cdf with λ = 6, lower = 0, upper = 3
   • P(X ≤ 3) = 0.151

3. (c) \(P(X \ge 7)\)

   • P(X ≥ 7) = 1 − P(X ≤ 6)
   • From the calculator: Poisson cdf with λ = 6, lower = 0, upper = 6
   • 1 − 0.6063 = 0.394

4. (d) \(P(X > 7)\)

   • P(X > 7) = 1 − P(X ≤ 7)
   • From the calculator: Poisson cdf with λ = 6, lower = 0, upper = 7
   • 1 − 0.7440 = 0.256

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

1. (a) The mean, E(X).

   • E(X) = λ = 11

2. (b) The variance, Var(X).

   • Var(X) = λ = 11

3. (c) The standard deviation.

   • standard deviation = √11 = 3.32

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.

   • X ~ Po(2.2)
   • From the calculator: Poisson pdf with λ = 2.2, x = 1
   • P(X = 1) = 0.244

2. (b) at least 3 cracks.

   • P(X ≥ 3) = 1 − P(X ≤ 2)
   • From the calculator: Poisson cdf with λ = 2.2, lower = 0, upper = 2
   • 1 − 0.6227 = 0.377

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.

   • X ~ Po(7.5)
   • E(X) = λ = 7.5

2. (b) The standard deviation of X.

   • Var(X) = 7.5
   • standard deviation = √7.5 = 2.74

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.

• 15 minutes is 1/4 of an hour, so λ = 24 ÷ 4 = 6
• X ~ Po(6)
• P(X < 6) = P(X ≤ 5)
• From the calculator: Poisson cdf with λ = 6, lower = 0, upper = 5
• P(X ≤ 5) = 0.446

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.

• 5 days: λ = 1.8 × 5 = 9
• X ~ Po(9)
• P(X ≥ 12) = 1 − P(X ≤ 11)
• From the calculator: Poisson cdf with λ = 9, lower = 0, upper = 11
• 1 − 0.8030 = 0.197

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.

• λ = 5 ÷ 4 = 1.25, so X ~ Po(1.25)
• P(2 ≤ X ≤ 4) = P(X ≤ 4) − P(X ≤ 1)
• From the calculator: Poisson cdf with λ = 1.25, lower = 2, upper = 4
• P(2 ≤ X ≤ 4) = 0.346