Simple variables and expectation
Section of Random variables and laws, part of Stochastic processes
Simple variables and expectation
| Type | Section |
| Taught in | Random variables and laws |
| Part | Stochastic processes |
A non-negative simple random variable takes finitely many non-negative values. Write
X=\sum_{i=1}^m a_i\, 1_{A_i}
with the A_i disjoint and measurable and a_i\ge 0. Define \mathbb{E} X=\sum_i a_i\mathrm{P}(A_i). For a general non-negative measurable X, choose simple X_n with 0\le X_n\uparrow X and define \mathbb{E} X=\lim_n\mathbb{E} X_n, allowing the value +\infty. The limit does not depend on the approximating sequence. This is the integral of X with respect to \mathrm{P}.
For a real-valued X, split X=X^+-X^- with X^+=\max(X,0) and X^-=\max(-X,0), so |X|=X^++X^-. If \mathbb{E}|X|<\infty, both parts have finite expectation and
\mathbb{E} X=\mathbb{E} X^+-\mathbb{E} X^-
is a finite, unambiguous number. If both parts have infinite expectation, the difference of the two infinities is not defined. The elementary formulas, a sum against a mass function and an integral against a density, are the special cases on Random variables and distributions. In general, when the integral exists,
\mathbb{E} g(X)=\int_\Omega g\bigl(X(\omega)\bigr)\,\mathrm{d}\mathrm{P}(\omega)=\int_{\mathbb{R}} g(x)\,\mathrm{d}F_X(x).
A mass function turns the second integral into a sum. A density f turns it into \int g(x)f(x)\,dx.
Used in
See also
Measurable random variable · Lebesgue decomposition · Kolmogorov extension theorem · Atom of a distribution