Editing Expected value (section)

===Expectations under convergence of random variables===
In general, it is not the case that <math>\operatorname{E}[X_n] \to \operatorname{E}[X]</math> even if <math>X_n\to X</math> pointwise. Thus, one cannot interchange limits and expectation, without additional conditions on the random variables. To see this, let <math>U</math> be a random variable distributed uniformly on <math>[0,1].</math> For <math>n\geq 1,</math> define a sequence of random variables
<math display="block">X_n = n \cdot \mathbf{1}\left\{ U \in \left(0,\tfrac{1}{n}\right)\right\},</math>
with <math>\mathbf{1}\{A\}</math> being the indicator function of the event <math>A.</math> Then, it follows that <math>X_n \to 0</math> pointwise. But, <math>\operatorname{E}[X_n] = n \cdot \Pr\left(U \in \left[ 0, \tfrac{1}{n}\right] \right) = n \cdot \tfrac{1}{n} = 1</math> for each <math>n.</math> Hence, <math>\lim_{n \to \infty} \operatorname{E}[X_n] = 1 \neq 0 = \operatorname{E}\left[ \lim_{n \to \infty} X_n \right].</math>

Analogously, for general sequence of random variables <math>\{ Y_n : n \geq 0\},</math> the expected value operator is not <math>\sigma</math>-additive, i.e.
<math display="block">\operatorname{E}\left[\sum^\infty_{n=0} Y_n\right] \neq \sum^\infty_{n=0}\operatorname{E}[Y_n].</math>

An example is easily obtained by setting <math>Y_0 = X_1</math> and <math>Y_n = X_{n+1} - X_n</math> for <math>n \geq 1,</math> where <math>X_n</math> is as in the previous example.

A number of convergence results specify exact conditions which allow one to interchange limits and expectations, as specified below.
* [[Monotone convergence theorem]]: Let <math>\{X_n : n \geq 0\}</math> be a sequence of random variables, with <math>0 \leq X_n \leq X_{n+1}</math> (a.s) for each <math>n \geq 0.</math> Furthermore, let <math>X_n \to X</math> pointwise. Then, the monotone convergence theorem states that <math>\lim_n\operatorname{E}[X_n]=\operatorname{E}[X].</math> {{pb}} Using the monotone convergence theorem, one can show that expectation indeed satisfies countable additivity for non-negative random variables. In particular, let <math>\{X_i\}_{i=0}^\infty</math> be non-negative random variables. It follows from the [[monotone convergence theorem]] that <math display="block">
\operatorname{E}\left[\sum^\infty_{i=0}X_i\right] = \sum^\infty_{i=0}\operatorname{E}[X_i].
</math>
* [[Fatou's lemma]]: Let <math>\{ X_n \geq 0 : n \geq 0\}</math> be a sequence of non-negative random variables. Fatou's lemma states that <math display="block">\operatorname{E}[\liminf_n X_n] \leq \liminf_n \operatorname{E}[X_n].</math> {{pb}} '''Corollary.''' Let <math>X_n \geq 0</math> with <math>\operatorname{E}[X_n] \leq C</math> for all <math>n \geq 0.</math> If <math>X_n \to X</math> (a.s), then <math>\operatorname{E}[X] \leq C.</math> {{pb}} '''Proof''' is by observing that <math display="inline"> X = \liminf_n X_n</math> (a.s.) and applying Fatou's lemma.
* [[Dominated convergence theorem]]: Let <math>\{X_n : n \geq 0 \}</math> be a sequence of random variables. If <math>X_n\to X</math> [[pointwise convergence|pointwise]] (a.s.), <math>|X_n|\leq Y \leq +\infty</math> (a.s.), and <math>\operatorname{E}[Y]<\infty.</math> Then, according to the dominated convergence theorem,
** <math>\operatorname{E}|X| \leq \operatorname{E}[Y] <\infty</math>;
** <math>\lim_n\operatorname{E}[X_n]=\operatorname{E}[X]</math>
** <math>\lim_n\operatorname{E}|X_n - X| = 0.</math>
* [[Uniform integrability]]: In some cases, the equality <math>\lim_n\operatorname{E}[X_n]=\operatorname{E}[\lim_n X_n]</math> holds when the sequence <math>\{X_n\}</math> is ''uniformly integrable.''