Poisson Distribution

A discrete random variable $X$ is said to have Poisson distribution with parameter $\lambda$ if its probability mass function is $$ \begin{equation*} P(X=x)= \left\{ \begin{array}{ll} \frac{e^{-\lambda}\lambda^x}{x!} , & \hbox{$x=0,1,2,\cdots; \lambda>0$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*} $$

Mean of Poisson Distribution

The expected value of Poisson random variable is

$E(X)=\lambda$.

Variance of Poisson Distribution

The variance of Poisson random variable is

$V(X) =\lambda$.

Moment Generating Funtion of Poisson Distribution

The moment generating function of Poisson distribution is

$M_X(t)=e^{\lambda(e^t-1)}$.

Probability generating function of Poisson Distribution

The probability generating function of Poisson distribution is

$P_X(t)=e^{\lambda(t-1)}$.

Mode of Poisson distribution

The condition for mode of Poisson distribution is

$\lambda-1 \leq x\leq \lambda$.

Poisson distribution as a limiting form of binomial distribution

In binomial distribution if $n\to \infty$, $p\to 0$ such that $np=\lambda$ (finite) then binomial distribution tends to Poisson distribution

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