Toolkit for stochastic processes
Section of Stochastic processes, part of Stochastic processes
Toolkit for stochastic processes
| Type | Section |
| Taught in | Stochastic processes |
| Part | Stochastic processes |
For integrable random variables, expectation adds whether or not the variables are independent:
\mathbb{E}(X+Y)=\mathbb{E} X+\mathbb{E} Y.
Independence is the extra assumption that makes variances add: \operatorname{Var}(X+Y)=\operatorname{Var}(X)+\operatorname{Var}(Y). The conditional versions separate two sources of variation. If the expectations exist,
\mathbb{E} X=\mathbb{E}\bigl\{\mathbb{E}(X\mid Y)\bigr\}, \qquad \operatorname{Var}(X)=\mathbb{E}\bigl\{\operatorname{Var}(X\mid Y)\bigr\}+\operatorname{Var}\bigl\{\mathbb{E}(X\mid Y)\bigr\}.
The first identity is the tower property. The second is the law of total variance: average the conditional variance, then add the variance of the conditional mean. Branching populations and random walks both use this split, because the next step is easy to describe once the present state is known.
For a non-negative random variable and a>0, Markov’s inequality is the comparison X\ge a\,1_{\{X\ge a\}}. Taking expectations gives a\mathbb{P}(X\ge a)\le\mathbb{E} X, so
\mathbb{P}(X\ge a)\le\frac{\mathbb{E} X}{a}.
If X has a finite variance, apply the same bound to (X-\mathbb{E} X)^2 at level \varepsilon^2. That is Chebyshev’s inequality:
\mathbb{P}\bigl(|X-\mathbb{E} X|\ge\varepsilon\bigr)\le\frac{\operatorname{Var}(X)}{\varepsilon^2}.
Both are bounds. They are useful when a mean or a variance is easy and the full distribution is not. They are the bounds that turn a finite variance into the weak law on Functions of random variables and limit theorems.
Used in
See also
Stochastic process · Stationary and independent increments · What a stochastic process models · State, clock, and assumptions · Modelling checklist for a stochastic process · What a model can identify · Markov’s inequality · Chebyshev’s inequality