Poisson Distribution Exam P Quick Reference

These flashcards cover key concepts, formulas, and properties related to the Poisson distribution as outlined in the lecture notes.

What is the Poisson distribution used to model?

The number of events in a fixed interval when events occur at a constant average rate, independently.

What is the notation for a Poisson distribution?

X ∼ Poisson(λ) where λ > 0 is the average rate of events.

What is the probability mass function (PMF) for the Poisson distribution?

P(X = k) = e^(-λ) λ^k / k! for k = 0, 1, 2, …

How do you calculate P(X = 0) in a Poisson distribution?

P(X = 0) = e^(-λ).

What does E[X] equal for a Poisson distribution?

E[X] = λ.

What is the variance of a Poisson distribution?

Var(X) = λ, which is equal to the mean.

How is the factorial moment E[X(X − 1)] calculated?

E[X(X − 1)] = λ^2.

What is the relationship between Poisson and Binomial distributions?

Poisson is an approximation of Binomial when n is large and p is small.

What does the Poisson process describe?

Events that arrive at a constant rate λ per unit of time.

Is the Poisson distribution memoryless?

No, only Exponential and Geometric distributions are memoryless.

What are the moments summary for a Poisson distribution?

E[X] = λ, E[X^2] = λ^2 + λ, E[X(X − 1)] = λ^2.

How do you find the median of a Poisson distribution?

There is no closed form; it is approximated by λ - 1/3 + 0.02λ.

What happens to the shape of the Poisson distribution as λ increases?

It becomes less skewed and approximately symmetric for λ ≥ 10.

If X ∼ Pois(λ1) and Y ∼ Pois(λ2) independently, what is X + Y?

X + Y ∼ Pois(λ1 + λ2).

How can Poisson distributions be used for conditional given sum problems?

If X ∼ Pois(λ) and Y ∼ Pois(μ) independent: X | (X + Y = n) ∼ Binomial(n, λ/(λ + μ)).

What is a key formula for the cumulative distribution function (CDF) of the Poisson distribution?

F(k) = Σ from i=0 to k of (e^(-λ) λ^i / i!).