Suppose $X$ is a random variable, then is it true to say that $$ E \left[ g \left( X_i \right) \right] = E\left[ g \left( X_j \right) \right] $$ ?
If so, why is that the case? Thanks a lot!
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$\begingroup$ What do the subscripts $i$ and $j$ represent? $\endgroup$– Alecos PapadopoulosCommented Nov 25, 2013 at 19:15
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$\begingroup$ @AlecosPapadopoulos Random sample. $\endgroup$– JohnKCommented Nov 25, 2013 at 19:18
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2$\begingroup$ Independence seems irrelevant here. If $X_i$ and $X_j$ are any random variables with the same distribution, then what needs to be shown is that $E[g(X_i)]=E[g(X_j)]$ for any measurable function $g$ (see "Law of the unconscious statistician"). $\endgroup$– whuber ♦Commented Nov 25, 2013 at 19:59
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1$\begingroup$ There is a subtlety that the Wikipedia article perhaps does not sufficiently emphasize: two random variables may differ but have the same distribution. For instance, let $X_1$ be the length of a chord of a unit circle whose endpoints are obtained from two independent directions uniformly distributed on $[0,2\pi)$. Let $X_2$ be same thing, but change its value to $-1$ on all diameters of the circle. These variables, although they differ on an infinite set, still have the same distribution. $\endgroup$– whuber ♦Commented Nov 25, 2013 at 20:15
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2$\begingroup$ The notation in your previous comment makes no sense: the value of a realization, such as "$0.5$", cannot have an expectation! You seem to be confusing the realizations with the random values used to model them. The value $0.5$ is a realization of a Uniform$[0,1]$ variable; likewise, $0.75$ is a realization of another Uniform$[0,1]$ variable. The variables--which are measurable real-valued functions defined on a sample space--have expectations, while their realizations are just numbers (which we anticipate will change in repetitions of the experiment). $\endgroup$– whuber ♦Commented Nov 25, 2013 at 20:38
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The random variables $X$ and $Y$ have the same distribution (meaning that $P(X\in A)=P(Y\in A)$ for every Borel set $A$), and you have a measurable function $g$. First, prove it when $g=I_A$, the indicator of some Borel set $A$. What are $\mathrm{E}[I_A(X)]$ and $\mathrm{E}[I_A(Y)]$? Give it a try.
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1$\begingroup$ I fear that my analysis knowledge does not cover Borel sets yet. Would you mind using beginner terminology? $\endgroup$– JohnKCommented Nov 25, 2013 at 23:07
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1$\begingroup$ Borel sets and measurable functions aside, the expected value of an Indicator Function is $P(X \in A)$, same with $Y$ which are equal. But how does this generalize to functions? $\endgroup$– JohnKCommented Nov 25, 2013 at 23:35
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1$\begingroup$ Sadly I do not for the time being. While I understand what the law of the unconsious statistician says, I do not yet see how it applies to the current situation. $\endgroup$– JohnKCommented Nov 25, 2013 at 23:42
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1$\begingroup$ Okay, we have then $E[g(X)]= \sum a_i P(X \in A_i) = \sum a_i P(Y \in A_i)$ , right? $\endgroup$– JohnKCommented Nov 26, 2013 at 0:05
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1$\begingroup$ I have so much to learn but statistics is fascinating. And you are very good at explaining. Thanks a lot. $\endgroup$– JohnKCommented Nov 26, 2013 at 0:30