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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.

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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.

MEAN

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

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

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

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

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

$$=\mu$$

VARIANCE

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

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

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

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

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

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

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

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  • $\begingroup$ 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 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 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 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 at 0:32

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