Mean and Variance of Poisson Distribution || Lesson 59 || Probability & Statistics | Learning Monkey

Mean and Variance of Poisson Distribution
In this class, We discuss Mean and Variance of Poisson Distribution.

The reader should have a prior knowledge og poisson distribution. Click Here.

Mean = µ = Expected Value

The below shows the complete derivation.
µ = ∑_(all x)▒〖x f(x; λ)〗
= ∑_(x=0)^∞▒x (e- λ λx)/x!
= e-λ ∑_(x=1)^∞ λx/(x-1)!
= λe-λ ∑_(x=1)^∞ λx-1/(x-1)!
= λe-λ eλ
= λ
µ = λ

Variance = σ2
= σ2 = ∑_(all x)▒x2f(x; λ) - µ2
= σ2 = E(x2) - (E(x))2
∑_(x=0)^∞▒x2(e-λ λx)/x!
= e-λ ∑_(x=0)^∞▒(x(x-1) + x) λx / x!
= e-λ ∑_(x=0)^∞▒(x(x-1) ) λx / x! + e-λ ∑_(x=0)^∞▒( x) λxe-λ / x!
Where e-λ ∑_(x=0)^∞▒( x) λxe-λ / x! = λ
= e-λλ2 ∑_(x=2)^∞▒( λx-2) / (x-2)! + λ
= e-λλ2eλ + λ
= λ2 + λ
E(x2) = λ2 + λ
= σ2 = λ2 + λ - λ2
= σ2 = λ

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