Question : A random sample $\displaystyle X_1,X_2,X_3 ..... X_n$ is taken. Show that the sample mean $\displaystyle \bar X$ is an unbiased estimate of the population mean $\displaystyle \mu$
You will find the proof in almost any Stats textbook and undoubtedly Google will turn up proofs in abundance. Also, see here:http://www.mathhelpforum.com/math-he...stimators.html (and no doubt using the Search tool will turn up more threads).
Since $\displaystyle f(x) = \frac{1}{ \sigma \sqrt{2 \pi}} . e^{- \frac{(x- \mu)^2}{2 \sigma}}$
it follows that
$\displaystyle ln f(x) = - ln \ \sigma \sqrt{2 \pi} - \frac{1}{2} \left( \frac{x- \mu}{\sigma} \right)^2
$
so that
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$\displaystyle = \frac{\sigma^2}{n}$..............Is the theorem u were talking about???