biased and unbiased

**wantanswers** · December 16th 2009, 07:50 AM

$<br /> S^2 = \frac{x-u}{n - 1} and S^2= \frac{x-u}{n }<br />$
I cant type the summation sign and x bar so i use u
Show that one is biased and unbiased

**matheagle** · December 16th 2009, 05:14 PM

what is u?
the sample mean is \bar X
the sum is \sum{i=1}^n X_i

are the X's from any particular distribution?

**novice** · December 17th 2009, 08:57 AM

Originally Posted by wantanswers

$<br /> S^2 = \frac{x-u}{n - 1} and S^2= \frac{x-u}{n }<br />$
I cant type the summation sign and x bar so i use u
Show that one is biased and unbiased

You mean is this:
$s^2=\frac{(X-\overline X)^2}{N}$ and $s^2=\frac{(X-\overline X)^2}{N-1}$

$\sigma^2= s^2=\frac{(X-\overline X)^2}{N}$ is the of the population variance and is unbiase, assuming that the population is infinite. It is like drawing balls from an urn with replacement, and it is also true according to the Central Limit Theorem.

On the other hand, the variance of the sample is not infinit, it is bias and just like drawing balls from an urn without replacement. Therefore, for a better estimate of the variance from a given sample, a correction factor $\frac{N}{N-1}$ is used.

The corrected estimate of sample variance becomes $(\frac{N}{N-1})\sigma^2= (\frac{N}{N-1})(\frac{(X-\overline X)^2}{N})$ , and the result is $s^2=\frac{(X-\overline X)^2}{N-1}$

**matheagle** · December 18th 2009, 04:49 PM

Is there a summation here? what is this U?.......
And I'm dying at ask 'what's a matta u'?
The Adventures of Rocky and Bullwinkle - Wikipedia, the free encyclopedia

The name of Bullwinkle's alma mater is Whassamatta U.
THis university was actually featured in the original cartoons,
though its exact location was never given; the film places it in Illinois.

Probably at my school?

**novice** · December 19th 2009, 09:20 AM

Originally Posted by matheagle

Is there a summation here? what is this U?.......
And I'm dying at ask 'what's a matta u'?
The Adventures of Rocky and Bullwinkle - Wikipedia, the free encyclopedia

The name of Bullwinkle's alma mater is Whassamatta U.
THis university was actually featured in the original cartoons,
though its exact location was never given; the film places it in Illinois.

Probably at my school?

Yah, there should be a $\Sigma$ .

Yah, hopeless. I should not have bothered.

**theodds** · December 20th 2009, 07:57 AM

Originally Posted by novice

$\sigma^2= s^2=\frac{(X-\overline X)^2}{N}$ is the of the population variance and is unbiase, assuming that the population is infinite.

That guy is biased for infinite populations. It is asymptotically unbiased though.

OP: You've left out at least some information, so I'll assume that $X_1, X_2, ... , X_n$ are iid, and that the terms are defined as Novice said above. Just take expectations, noting that $\sum _{i = 0} ^ n \left( X_i - \bar{X} \right) ^2 = \sum _{i = 0} ^ n X_i ^2 - n\bar{X}^2$ . You'll have to flip around the formula $\mbox{Var} X = EX^2 - (EX)^2$ , but it isn't more than a few lines.

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