Gamma Distribution - lec. 16

ucla | MATH 170E | 2023-02-16T21:26

---

Table of Contents

• Definitions
• Big Ideas
  • Gamma R.V.
    • PDF (@import url(‘https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css’)$\alpha \in \text{Z}\ge 1$)
    • Gamma function
    • MGF
    • Mean and Variance
• Resources

Definitions

---

Big Ideas

---

Gamma R.V.

• given Poisson $\lambda >0$ and if $\alpha \in \text{Z}\ge 1$ let $X$ be the time of the $\alpha$th arrival
• $X\sim \text{Gamma}(\alpha ,\theta )$ s.t. $\alpha =1⟹X\sim \text{Exp}(\theta =\frac{1}{\lambda })$

PDF (@import url(‘https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.2/katex.min.css’)$\alpha \in \text{Z}\ge 1$)

$f_X(x)=\frac{1}{{\theta }^{\alpha }(\alpha -1)!}{x}^{\alpha -1}{e}^{-x/\theta }x>0$

$f_X(x)=\frac{1}{{\theta }^{\alpha }\mathrm{\Gamma }(\alpha )}{x}^{\alpha -1}{e}^{-x/\theta }x>0$

Gamma function

$\mathrm{\Gamma }(\alpha )={\int }_0^{\text{infin}}{x}^{\alpha -1}{e}^{-x}dx\alpha >0$

$\mathrm{\Gamma }(1)=1$

$\mathrm{\Gamma }(\alpha )=(\alpha -1)\mathrm{\Gamma }(\alpha -1)\alpha >1$

$\mathrm{\Gamma }(\alpha )=(\alpha -1)!\alpha \ge 1$

MGF

$M_X(t)=\frac{1}{(1-\theta t{)}^{\alpha }}t<\frac{1}{\theta }$

Mean and Variance

$\mathbb E[X]=\alpha\theta\space$

Resources

---

📌

**SUMMARY
**