Further Stats Chapter 2: Poisson Distributions

8 flashcards · Mathematics · by TheoTyson

1. What is the formula for the Poisson distribution X∼Po⁡(λ)X \sim {\operatorname {Po}} ( \lambda)X∼Po(λ)?

   P(X=x)=P (X = x) =P(X=x)= e−λλxx!\frac{e^{- \lambda} \lambda^x}{x!}x!e−λλx​ x=0,1,2,3...x = 0, 1, 2 , 3 ...x=0,1,2,3...

2. What are the criteria for a Poisson distribution?

   The events must be independent. They must occur singly, in space or time. At a constant average rate.

3. How do you add two poisson distributions X∼Po⁡(λ)X \sim {\operatorname {Po}} (\lambda)X∼Po(λ) and Y∼Po⁡(μ)Y \sim {\operatorname {Po}} (\mu)Y∼Po(μ) ?

   X+Y∼Po⁡(λ+μ)X + Y \sim {\operatorname {Po}} (\lambda + \mu)X+Y∼Po(λ+μ)

4. What is the mean of X∼Po⁡(λ)X \sim {\operatorname {Po}} (\lambda)X∼Po(λ) ?

   E(X)=λE(X) = \lambdaE(X)=λ

5. What is the variance of X∼Po⁡(λ)X \sim {\operatorname {Po}} (\lambda)X∼Po(λ) ?

   Var⁡(X){\operatorname{Var}} (X)Var(X) = σ2\sigma ^ 2σ2 = λ\lambdaλ

6. What is the mean of X∼B⁡(n, p)X \sim {\operatorname {B}} (n, \, p)X∼B(n,p) ?

   E⁡(X)=μ=np{\operatorname {E}} (X) = \mu = npE(X)=μ=np

7. What is the variance of X∼B⁡(n, p)X \sim {\operatorname {B}} (n, \, p)X∼B(n,p) ?

   Var⁡(X){\operatorname {Var}} (X)Var(X) = σ2\sigma ^ 2σ2 === np(1−p)np(1-p)np(1−p)

8. When can the distribution X∼B(n,p)X \sim \text{B} (n, p)X∼B(n,p) be approximated to the distribution Y∼Po(λ)Y \sim \text{Po}(\lambda)Y∼Po(λ) ?

   If nnn is large and ppp is small, then λ∼np\lambda \sim npλ∼np .