From a sample we have the expected value: $E[x] = μ$.

And if we have differents trials $n=654$ the mean still is $E[x] = μ$.

What is the reason of this relation? What causes this relationship to be true?


1 Answer 1


It depends what you mean when you say from the different trials the mean is the same. Presumably you're describing the statistic that corresponds to the mean of the trials (since I can make any function of the trials and have it not equal the mean of a single trial!)

Consider $S_n = \frac{1}{n} \sum_{j=1}^n X_j$, where the $X_j$ are the results of your trial, each independent and with mean $E(X_j) = \mu$. Then one can show (by linearity of the expectation operator): $$ E(S_n) = E \left(\frac{1}{n} \sum_{j=1}^n X_j \right) = \frac{1}{n} \sum_{j=1}^n E(X_j) = \frac{1}{n} \sum_{j=1}^n \mu = \mu. $$


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