gamma distribution

p.d.f

$\Gamma(\alpha,\beta)=\frac{1}{\Gamma(\alpha)}x^{\alpha-1}\beta^{\alpha}e^{-\beta x}$ $x>0$

k-th moment

$$
\begin{equation}

E[X^k]=\int_0^{\infty} \frac{1}{\Gamma(\alpha)}x^{\alpha-1}x^{k}\beta^{\alpha}e^{-\beta x}=\frac{\Gamma(k+\alpha)}{\Gamma(\alpha)}\frac{1}{\beta^k}

\end{equation}
$$

So, we can get $E[X]=\frac{\alpha}{\beta}$ and $Var(X)=\frac{\alpha}{\beta^2}$

moment generating function

$\left( 1-\frac{t}{\beta}\right)^{-\alpha}$ $t<\beta$

characteristic function

$\left( 1-\frac{it}{\beta}\right)^{-\alpha}$

Using the properties of characteristic we can get the properties as follows

properties

  1. If $X_1 \sim \Gamma(\alpha_1,\beta)$ and $X_2 \sim \Gamma(\alpha_2, \beta)$, then $X_1+X_2 \sim \Gamma(\alpha_1+\alpha_2,\beta)$
  2. If $X \sim \Gamma(\alpha,\beta)$, then $kX \sim \Gamma(\alpha,\frac{\beta}{k})$
  3. Exponential distribution $Exp(\lambda)\sim \Gamma(1,\lambda)$ and Chi square distribution $\chi^2(n)\sim \Gamma(\frac{n}{2},\frac12)$

poisson distribution

p.m.f (probability mass function)

$P(X=x)=\frac{\lambda^x}{x!}e^{-\lambda x}$ $x\in \mathbb{N}$

characteristic function

$X\sim P(\lambda)$

$$
\begin{equation}

E[e^{itX}]=\sum_{k=0}^{\infty}e^{itx}\frac{\lambda^x}{x!}e^{-\lambda x}=e^{-\lambda}e^{\lambda e^{it}}\sum_{k=0}^{\infty}

\frac{(e^it\lambda)^x}{x!}e^{-\lambda e^it}=e^{\lambda(e^{it}-1)}

\end{equation}
$$
Using the properties of characteristic we can get the property as follows

property

If $ X_1, X_2 \dots X_n $ i.i.d $\sim P(\lambda)$, then $\sum_{i=1}^{n}X_i \sim P(n\lambda)$