Suppose we have a distribution that has some pdf, say, $f(x)= 4(3-x^3)$ for $0<x<2$, and $0$ elsewhere. I am able to find $E(X)$ and $Var(X)$ using the standard definition of expected value (i.e. integral of $xf(x)$) and variance. But suppose we are given that a random sample of some size, say $15$, has been obtained from this distribution, and so this sample itself will have a mean, and the problem is to find the expected value of the sample mean. I initially thought that the expected value and variance of the sample mean will just be $E(X)$ and $Var(X)$ respectively, but I am not sure if the sample mean and variance will be influenced by the sample size.

My query is whether the expected value of the sample mean is the same as $E(X)$ regardless of sample size, and if not what is the approach to determining the expected value of the sample mean taking into account the sample size.

Thank you.


Make the assumption that $X_1,\dots,X_n\overset{iid}{\sim} \text{Distribution}$ have a mean $\mu$ and a variance $\sigma^2$. The $\text{Distribution}$ need not be normal (though it could be).

I've given the full derivations. My suggestion is to scroll down one line at a time and try to anticipate my next line so that you can prove it yourself.


$$\mathbb{E}\big[\bar{X}\big] =\mathbb{E}\Bigg[\dfrac{1}{n}\sum_{i=1}^n X_i\Bigg]$$

$$=\dfrac{1}{n}\mathbb{E}\Bigg[\sum_{i=1}^n X_i\Bigg]$$

$$=\dfrac{1}{n}\sum_{i=1}^n\mathbb{E}\big[ X_i\big]$$

$$= \dfrac{1}{n}\sum_{i=1}^n \mu$$

$$=\dfrac{1}{n} n\mu$$



$$Var\big(\bar{X}\big) = Var\Bigg(\dfrac{1}{n}\sum_{i=1}^n X_i\Bigg)$$

$$=\dfrac{1}{n^2}Var\Bigg(\sum_{i=1}^n X_i\Bigg) $$

$$=\dfrac{1}{n^2}\sum_{i=1}^n Var\big(X_i\big) $$

$$=\dfrac{1}{n^2}\sum_{i=1}^n \sigma^2 $$

$$=\dfrac{1}{n^2}n \sigma^2 $$

$$=\dfrac{\sigma^2}{n} $$

NOTE: This is NOT a proof of the central limit theorem.

  • $\begingroup$ Thank you very much, this is very helpful. I am new to this site, could I ask how to write the mathematical notation? (I have looked at the menu bar in the question box, where there are a few options (eg 'Code', HTML' etc) but there did not seem to be one for an equation editor) $\endgroup$
    – Joanne
    Oct 29 '20 at 9:33
  • $\begingroup$ Throw dollar signs around the mathematical text and use the usual LaTeX notation. We have a tour that newcomers to Cross Validated can take that might be worth taking, too. $\endgroup$
    – Dave
    Oct 29 '20 at 9:54
  • $\begingroup$ sure, but where exactly can I find the LaTex tool on the question bar? I have tried all the icons on the question box menu bar (and the tour also did not specify this) $\endgroup$
    – Joanne
    Oct 30 '20 at 0:19
  • $\begingroup$ There is no LaTeX tool. You put text like \eta^{\xi}=\int_{-\pi}^{\pi}e^{-x^2}dx in between dollar signs. If you have further questions about LaTeX text on here, please visit our meta. $\endgroup$
    – Dave
    Oct 30 '20 at 0:32

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