variable

$$\mathrm{Normal}(\mu, \sigma^2)$$

pdf

$$f(z) = \frac{1}{\sqrt{2 \pi \sigma^2}} \ \exp \left( -\frac{(z-\mu)^2}{2 \sigma^2} \right)$$

mean

$$\mu$$

variance

$$\sigma^2$$

mgf

$$m(t) = \exp \left( \mu t + \frac{\sigma^2 t^2}{2} \right)$$

variable

$$\mathrm{T}_\nu$$

pdf

$$f(z) = \frac{\Gamma \left( \frac{\nu+1}{2} \right)}{\sqrt{\nu \pi} \ \Gamma \left( \frac{\nu}{2} \right)} \ \left( 1 + \frac{z^2}{\nu} \right)^{-\frac{\nu+1}{2}}$$

mean

$$0$$

variance

$$\frac{\nu}{\nu-2}$$

mgf

$$E[X^{2n}] = \frac{\Gamma \left( \frac{\nu + 1}{2} \right) \ \Gamma \left( \nu - n \right)}{\sqrt{\pi} \ \Gamma \left( \frac{\nu}{2} \right)}$$