Gamma distribution

Definition: $X\sim Gamma(\alpha ,\lambda )$

• Continuous random variable
• For $a>0$ and $\lambda >0$
• The gamma distribution with $\lambda =\frac{1}{2}$ and α = n/2 is called the ${\chi }_n^2$ distribution (Chi-squared distribution) with n degrees of freedom
• $X\sim Gamma(n,\lambda )$ Gamma distribution denotes the amount of time one has to wait until a total of $n$ events has occurred knowing avgly $\lambda$ events occurs
• $\lambda$ is the frequency of events happening, how many events per time frame
• Nb of events happened $\sim Po(\lambda t)$
• $X\sim Gam(1,\lambda )=X\sim Exp(\lambda )\to$When $n=1$, its the same as Exponential Distribution

Probability density function:

• $f(x)=\frac{\lambda e^{-\lambda x}(\lambda x{)}^{\alpha -1}}{\Gamma (\alpha )}\text{for }x\ge 0\text{and}0\text{for }x<0$
• Where $\Gamma (\alpha )$ is Gamma function

Gamma function:

• Defined as:$\Gamma (\alpha )={\int }_0^{\infty }{e}^{-y}{y}^{\alpha -1}dy$
  • When $\alpha =n\in {\mathbb{Z}}^{+}$:$\Gamma (n)=(n-1)!$

Expected value $E[X]=\frac{\alpha }{\lambda }$

Variance $Var(X)=\frac{\alpha }{{\lambda }^2}$

Sum of independent Gamma variables:

• If $X\sim Gam(s,\lambda )$ and $Y\sim Gam(t,\lambda )$, then $X+Y\sim (s+t,\lambda )$