Contents

Expectations¶

Given an random variable \(X\) with PDF \(f(x)\) we define its expected value (mean) as:

\[\begin{split} E[X] = \int_{-\infty}^{\infty} xp(x)dx \\ \end{split}\]

Functions of random variables¶

Given a function \(g: R \rightarrow R\), \(g(X)\) is also an random variable we define its expectation as:

\[\begin{split} E[g(X)] = \int_{-\infty}^{\infty} g(x)p(x)dx \\ = E_{x \sim p(x)}[g(x)] \end{split}\]

This is the weighted average of \(g\). In the special case when \(g\) is the identity function we get mean.

Properties¶

  • \(E[a] = a\) for any constant \(a\)

  • \(E[af(X)] = a E[f(X)]\) for any constant \(a\)

  • \(E[af(X) + bg(X)] = aE[f(X)] + bE[g(X)]\) Linearity

Variance¶

Variance of a random variable X is a measure how concentrated the distribution of a random variable X is around its mean.

\[ Var[X] = E[(X - E[X])^2] = E[X^2] - E[X]^2 \]

Units of variance are the squared units of the original.

Properties¶

  • \(Var[a] = 0\) for any constant

  • \(Var[f(x) + a] = Var[f(x)]\) for any constant \(a\), thus shifting an random variable has no effect on its variance.

  • \(Var[af(X)] = a^2Var[f(X)]\) for any constant \(a\). Variance is measured in units squared thus any constant is also squared

  • \(Var[f(x) + g(Y)] = Var[f(Y)] + Var[g(Y)]\) if \(X \perp Y\)

Standard deviation¶

We take the square root of the variance. $\( std[X] = \sqrt{Var[X]} \)$

Standard deviation has the same units as the original

Skew¶

Curtosis¶

Discrete random variables¶

More or less the same but we replace the integration with a sum over the domain of a random variable.

Rules Determinants Adam (adaptive moment estimation method )