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# Probability

## Parametric Distributions

Many common distributions depend on specific parameters. Parameters are frequently classified into one of several possible types:

• location parameter: shifts the distribution e.g. Gaussian mean
• scale (inverse rate): stretches/squeezes the distribution e.g. Laplace diversity
• shape: changes the shape e.g. Beta $$\alpha, \beta$$

Some common discrete, continuous and special parametric distributions are:

## Distances and Divergences

Probability distances and divergences have commonly encountered properties. Some common probability distances and divergences are

### Probability Integral Transform

Theorem: For any random variable $$X$$, its CDF $$F_X(x)$$ is distributed uniformly over $$(0,1)$$. That is, if we define $$Y = F_X(x)$$, then $$Y \sim \mathcal{U}(0,1)$$.

Proof: \begin{align*} P(Y \leq y) &= P(F_X(x) \leq y)\\ &= P(x \leq F_X^{-1}(y))\\ &= F_X(F_X^{-1}(y))\\ &= y \end{align*} Since only\mathcal{U}(0,1)$$has a CDF$$F_Y(y) = P(Y \leq y) = y$$, we conclude that$$Y is distributed uniformly.