01 Distribution of values

Specification and classes

A distribution is often specified by a pdf or a cdf involving certain parameters. Or it may be specified by a stochastic process generating some values: ie in terms of other other distributions.

Sometimes, the density specified need not even be proper (sum/ integrate to 1) to be useful: Eg: In applying the conditional probability inversion trick.

Notation

If the pdf of $X$ is a member (identified by the parameter $p_1$) of the function family $\{f(p)\}$, we write $X\sim f(p_1)$.

Parameter types

Location parameters specify important points in the distribution: Eg: $\mu$ in $N(\mu ,{\sigma }^2)$ distribution.

Scale parameters specify how spread-out the distribution is. A parameter $s$ is a scale parameter if, having set the location parameter to 0, $CDF(x;ks)=CDF(x/k;s)$ Eg: $h$ in $C(x;\mu ,h)$ distribution, and $\sigma$ in $N(\mu ,{\sigma }^2)$.

All other parameters are called shape parameters.

Specify continuous distribution over bounded support

Take Indicator fn $I_{(a,b)}$: See algebra ref. So, if U(a,b): $f(x)=(b-a{)}^{-1}{I}_{(a,b)}(x)$. Not differentiable in boundaries.

Inference, Sampling from distribution

See randomized algorithms ref.