Variance of poisson distribution

Answer

Variance of Poisson Distribution

The formula for mean OR the expected value of a poisson distribution is $\lambda$. 

The formula of Variance is;

$Var \left(x\right)=E\left[\right(X-E{\left(X\right)}^2]$

$=E\left[X^2\right]-E{\left[X\right]}^2$

Thus the Poisson Distribution Variance will be;

$Var \left(x\right)=M_x^{II}\left(t\right) - (M_x^I{\left(t\right))}^2$

$M_x^I\left(t\right)=\lambda e^t{e}^{\lambda \left(e^{t-1}\right)} + \lambda e^t\lambda e^t{e}^{\lambda (e^t-1)}$

Upon replacing t with 0 we have;

$M_x^{II}\left(0\right)=\lambda e^0{e}^{\lambda \left(e^{0-1}\right)} + \lambda e^0\lambda e^0{e}^{\lambda (e^0-1)}$

$M_x^{II}\left(0\right)=\lambda +{\lambda }^2$

$Var \left(x\right)=M_x^{II}\left(0\right) - (M_x^I{\left(0\right))}^2$

$=\lambda +{\lambda }^2-{\left(\lambda \right)}^2$

$=\lambda$

The Variance of Poisson Distribution is $\lambda$

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