Poisson distribution practice questions.

The number of emails arriving per minute follows a Poisson distribution with mean 2. Calculate the probability that exactly 3 emails arrive in a given minute, to 3 s.f.

• 0.271
• 0.180
• 0.135
• 0.090

Answer: 0.180

P(X = 3) = e^(−2) × 2³/3! = e^(−2) × 8/6 ≈ 0.180.

Cars pass a checkpoint at an average rate of 3 per minute, following a Poisson distribution. Calculate the probability that no cars pass in a given minute, to 3 s.f.

• 0.149
• 0.0224
• 0.0498
• 0.0996

Answer: 0.0498

P(X = 0) = e^(−3) ≈ 0.0498.

Two independent Poisson variables have means 4 and 6. State the distribution of their sum.

• Poisson with mean 10
• Poisson with mean 24
• Poisson with mean 5
• Normal with mean 10

Answer: Poisson with mean 10

The sum of independent Poisson variables is Poisson with mean equal to the sum of the means, so Poisson with mean 10.

Accidents at a junction occur at an average rate of 2 per week, following a Poisson distribution. Calculate the probability of at least one accident in a week, to 3 s.f.

• 0.135
• 0.271
• 0.594
• 0.865

Answer: 0.865

P(X ≥ 1) = 1 − P(X = 0) = 1 − e^(−2) ≈ 0.865.

Defects occur on a production line at an average rate of 3 per day (Poisson). State the mean number of defects expected over a 2-day period.

• 3
• 6
• 9
• 1.5

Answer: 6

The rate scales with the interval, so over 2 days the mean is 3 × 2 = 6.

A rare fault occurs at an average rate of 0.5 per hour, following a Poisson distribution. Calculate the probability of exactly one fault in an hour, to 3 s.f.

• 0.607
• 0.152
• 0.303
• 0.500

Answer: 0.303

P(X = 1) = e^(−0.5) × 0.5 ≈ 0.303.