Poisson Random Variable : Exercises
Introduction
Example problems
Click on the problems to reveal the solution
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. $$ \mathbb{E}(X^2) = \frac{\textrm{d}^2M_X(0)}{\textrm{d}t^2} $$ The second derivative for the moment generating function of the Poisson random variable is given by: $$ \frac{\textrm{d}^2M_X(t)}{\textrm{d}t^2} = \lambda e^t e^{\lambda(e^t - 1)} + \lambda^2 e^{2t} e^{\lambda(e^t - 1)} $$ At $t=0$ then $\mathbb{E}(X^2) = \lambda e^0 e^{\lambda(e^0 - 1)} + \lambda^2 e^{0} e^{\lambda(e^0 - 1)} = \lambda + \lambda^2$ The variance is thus $\textrm{var}(X) = \mathbb{E}(X^2) - [\mathbb{E}(X)]^2 = \lambda + \lambda^2 - \lambda^2 = \lambda$