Distributions

Lognormal Distribution

• Some phenomena grow multiplicatively rather than additively. If
  $W\sim \mathcal{N}(\mu ,{\sigma }^2),X=\exp (W),$
  then $X$ is lognormal.
• PDF $$f(x)=\frac{1}{x\sigma \sqrt{2\pi }}\exp [-\frac{(\ln x-\mu {)}^2}{2{\sigma }^2}],x>0.$$
• Mean and variance $$\mathbb{E}[X]=e^{\mu +\frac{1}{2}{\sigma }^2},\mathrm{Var}[X]=(e^{{\sigma }^2}-1)e^{2\mu +{\sigma }^2}.$$

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Gamma Distribution

• The gamma function generalises the factorial: $$\Gamma (r)={\int }_0^{\infty }{x}^{r-1}{e}^{-x}dx,r>0,\Gamma (r)=(r-1)\Gamma (r-1).$$
• With shape $k>0$ and rate $\lambda >0$, the gamma PDF is $$f(x)=\frac{{\lambda }^k}{\Gamma (k)}{x}^{k-1}{e}^{-\lambda x},x>0.$$
• Mean and variance $$\mu =\frac{k}{\lambda },{\sigma }^2=\frac{k}{{\lambda }^2}.$$ (For integer $k$ the gamma reduces to an Erlang; for $k=\frac{n}{2}$ and $\lambda =\frac{1}{2}$ it produces the ${\chi }^2$ distribution.)

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Beta Distribution

• A continuous distribution bounded on $[0,1]$; useful for proportions.
• PDF with shape parameters $\alpha >0,\beta >0$: $$f(x)=\frac{\Gamma (\alpha +\beta )}{\Gamma (\alpha )\Gamma (\beta )}{x}^{\alpha -1}(1-x{)}^{\beta -1},0<x<1.$$
• Tuning $\alpha$ and $\beta$ changes the shape:
  • $\alpha =\beta =1$ → uniform;
  • $\alpha =\beta$ → symmetric about $x=0.5$;
  • $\alpha <1$ or $\beta <1$ → modes at the boundaries.
• Mean and variance $$\mu =\frac{\alpha }{\alpha +\beta },{\sigma }^2=\frac{\alpha \beta }{(\alpha +\beta {)}^2(\alpha +\beta +1)}.$$

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