Poisson Random Variable : Exercises
Introduction
Example problems
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Problem 1
Problem 2
To calculate the variance we need to first caclulate the second moment of the distribution. We can find this quantity from the moment generating function by finding the value of the second derivative at t=0. i.e. E(X2)=d2MX(0)dt2 The second derivative for the moment generating function of the Poisson random variable is given by: d2MX(t)dt2=λeteλ(et−1)+λ2e2teλ(et−1) At t=0 then E(X2)=λe0eλ(e0−1)+λ2e0eλ(e0−1)=λ+λ2 The variance is thus var(X)=E(X2)−[E(X)]2=λ+λ2−λ2=λ