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November 10th, 2017, 04:53 AM   #1
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proof

How I can prove the sample mean of "n" samples is the unbiased estimator of the population mean.

SHAZAM Unbiased Estimation
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Last edited by MATHEMATICIAN; November 10th, 2017 at 05:03 AM.
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November 10th, 2017, 05:05 AM   #2
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$\begin{align*}
&E[S_n] = \\

&E\left[\dfrac 1 n \sum \limits_{k=1}^n~X_k\right] = \\

&\dfrac 1 n \sum \limits_{k=1}^n E[X_k]

\end{align*}$

Assuming your samples all come from the same distribution

$E[X_k] = \mu,~\forall k \in \{1,2, \dots n\}$

so

$E[S_n] = \dfrac 1 n (n \mu) = \mu$
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November 10th, 2017, 07:38 AM   #3
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Quote:
Originally Posted by romsek View Post
$\begin{align*}
&E[S_n] = \\

&E\left[\dfrac 1 n \sum \limits_{k=1}^n~X_k\right] = \\

&\dfrac 1 n \sum \limits_{k=1}^n E[X_k]

\end{align*}$

Assuming your samples all come from the same distribution

$E[X_k] = \mu,~\forall k \in \{1,2, \dots n\}$

so

$E[S_n] = \dfrac 1 n (n \mu) = \mu$
too advanced for me

can you please explain it equation by equation in simple words?
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November 10th, 2017, 09:50 AM   #4
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"SHAZAM" sounds like a secret word in Ali-Baba-like old stories !!
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November 11th, 2017, 07:29 AM   #5
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Quote:
Originally Posted by Denis View Post
"SHAZAM" sounds like a secret word in Ali-Baba-like old stories !!
DENIS could you please explain me those equations?
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November 11th, 2017, 09:51 AM   #6
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I don't know how you can use language like "sample mean" and not understand the notation below.

You know the sample mean is the sum of say $n$ samples divided by $n$.

$S_n$ represents this sample mean, and we want to find the expectation of it.

That is what

$E[S_n] = E\left[\dfrac 1 n \sum \limits_{k=1}^n~X_k\right]$

says.

Then due to properties of expectation we can push the expectation into the sum to obtain

$E[S_n] = \dfrac 1 n \sum \limits_{k=1}^n E[X_k]$

You are told you are sampling from a single population with a given mean $E[X_k] = \mu,~\forall k$

so substituting this in we get

$E[S_n] = \dfrac 1 n \sum \limits_{k=1}^n \mu = \dfrac 1 n (n \mu) = \mu$

I don't mean to be rude but if you can't understand the above you need to do some more reading.
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