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

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

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

   • From the calculator: Poisson pdf with λ = 3.4, x = 1
   • P(X = 1) = 0.113

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

   • From the calculator: Poisson pdf with λ = 3.4, x = 2
   • P(X = 2) = 0.193

3. (c) \(P(X = 6)\)

   • From the calculator: Poisson pdf with λ = 3.4, x = 6
   • P(X = 6) = 0.0716

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

1. (a) \(P(X \le 5)\)

   • From the calculator: Poisson cdf with λ = 7.5, lower = 0, upper = 5
   • P(X ≤ 5) = 0.241

2. (b) \(P(X < 5)\)

   • Fewer than 5 means at most 4
   • From the calculator: Poisson cdf with λ = 7.5, lower = 0, upper = 4
   • P(X ≤ 4) = 0.132

3. (c) \(P(X \ge 9)\)

   • P(X ≥ 9) = 1 − P(X ≤ 8)
   • From the calculator: Poisson cdf with λ = 7.5, lower = 0, upper = 8
   • 1 − 0.6620 = 0.338

4. (d) \(P(X > 9)\)

   • P(X > 9) = 1 − P(X ≤ 9)
   • From the calculator: Poisson cdf with λ = 7.5, lower = 0, upper = 9
   • 1 − 0.7764 = 0.224

\(X \sim \text{Po}(13)\). Write down the mean of X, then find the variance and the standard deviation of X.

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

   • E(X) = λ = 13

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

   • Var(X) = λ = 13

3. (c) The standard deviation.

   • standard deviation = √13 = 3.61

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.

   • X ~ Po(3.1)
   • From the calculator: Poisson pdf with λ = 3.1, x = 4
   • P(X = 4) = 0.173

2. (b) at least 2 flaws.

   • P(X ≥ 2) = 1 − P(X ≤ 1)
   • From the calculator: Poisson cdf with λ = 3.1, lower = 0, upper = 1
   • 1 − 0.1847 = 0.815

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.

   • X ~ Po(5.5)
   • E(X) = λ = 5.5

2. (b) The standard deviation of X.

   • Var(X) = 5.5
   • standard deviation = √5.5 = 2.35

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.

• 10 minutes is 1/6 of an hour, so λ = 18 ÷ 6 = 3
• X ~ Po(3)
• P(X < 4) = P(X ≤ 3)
• From the calculator: Poisson cdf with λ = 3, lower = 0, upper = 3
• P(X ≤ 3) = 0.647

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.

• 6 hours: λ = 0.7 × 6 = 4.2
• X ~ Po(4.2)
• P(X > 5) = 1 − P(X ≤ 5)
• From the calculator: Poisson cdf with λ = 4.2, lower = 0, upper = 5
• 1 − 0.7531 = 0.247

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.

• λ = 9 × 2 ÷ 6 = 3, so X ~ Po(3)
• P(2 ≤ X ≤ 5) = P(X ≤ 5) − P(X ≤ 1)
• From the calculator: Poisson cdf with λ = 3, lower = 2, upper = 5
• P(2 ≤ X ≤ 5) = 0.717