What is an example of a sequence $X_1,X_2,...$ such that $X_n\rightarrow X$ in probability, but $\mathbb{E}(X_n)$ does not converge to $\mathbb{E}(X)$?
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Consider for instance: $$ X_n = \begin{cases} n \text { with probability }\frac 1n\\ 0 \text { with probability }1 - \frac 1n\\ \end{cases} $$
- $P(|X_n| > \epsilon) = \frac 1n\to 0$ hence $X_n\to X = 0$ in probability.
- $EX_n = \frac 1n\times n + \left(1-\frac 1n\right)\times 0 = 1 \to 1 \neq EX$
mookid
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But how do i prove that this converges in probability but the expectation doesn't? – Apr 28 '15 at 19:58
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the exprectation isn't a problem... for the convergence, go back to the definition? – mookid Apr 28 '15 at 20:00
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Why isnt the expectation a problem? – Apr 28 '15 at 20:01
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1@TonyStrong I edited with more details. – mookid Apr 28 '15 at 20:03