The Poisson distribution

The queue that never ends

• Customers arrive at a shop at an average rate of 3 per minute. What's the probability that exactly 5 arrive in the next minute?
• The Poisson distribution 泊松分布 models rare events happening at a steady rate — from phone calls to radioactive decay.

The Poisson distribution

• $X \sim \text{Po}(\lambda)$ models the number of random events in a fixed interval 区间 at a steady rate $\lambda$:

People arriving at random in a queue follow a Poisson distribution

Properties of the Poisson

• For a Poisson variable, the mean 平均数 and variance 方差 both equal $\lambda$: $E(X) = \text{Var}(X) = \lambda$.
• This is unique — no other common distribution has this property.

For Po(3) the mean and variance are both 3; the bars give P(X = k).

Poisson approximation to the binomial

• The Poisson approximates the binomial when $n$ is large and $p$ is small.
• Use $\lambda = np$ as the Poisson parameter 参数.
• Example: $B(200, 0.01) \approx \text{Po}(2)$.

Worked example — approximation

• $X \sim B(100, 0.02)$. Approximate with Poisson: $\lambda = np = 100 \times 0.02 = 2$.
• $P(X = 0) \approx e^{-2} \dfrac{2^0}{0!} = e^{-2} = 0.1353$.
• The normal distribution approximates the binomial distribution (with a continuity correction) when appropriate.

You've got it

Write from a Poisson mean of 5

• Independent Poisson variables have means 2 and 3.
• Find the mean and variance of their sum.