Binomial distribution proof of mean

**eugene1687** · December 6th 2007, 06:36 AM

HELP PLEASE...
Can anyone provide a proof for the variance of binomial distribution?
Show that it is npq without using the Bernoulli distribution and independence way..( which is the typical way of summations or expectations)
please help me..
thank you so much..

**ThePerfectHacker** · December 6th 2007, 07:56 AM

Originally Posted by eugene1687

HELP PLEASE...
Can anyone provide a proof for the variance of binomial distribution?
Show that it is npq without using the Bernoulli distribution and independence way..( which is the typical way of summations or expectations)
please help me..
thank you so much..

Here is the solution from my probability book.

$E[X^k] = \sum_{i=0}^n i^k {n\choose i} p^i(1-p)^{n-i} = \sum_{i=1}^n i^k {n\choose i}p^i (1-p)^{n-i}$
Use the identity,
$i{n\choose i} = n{{n-1}\choose {i-1}}$
Thus, letting $j=i-1$ ,
$E[X^k] = np\sum_{j=0}^{n-1}(j+1)^{k-1}{{n-1}\choose j}p^j(1-p)^{n-1-j}=$ $npE[(Y+1)^{k-1}] \mbox{ where }Y \sim \mbox{bino}(n-1,p)$ .

So that means,
$E[X] = np E[(Y+1)^0] = np$
$E[X^2] = np E[(Y+1)^1] = np[(n-1)p+1]$
That means,
$\mbox{Var}[X] = E[X^2] - (E[X])^2 = np(1-p)$

**Plato** · December 6th 2007, 08:00 AM

Originally Posted by eugene1687

HELP PLEASE...
Can anyone provide a proof for the variance of binomial distribution?
Show that it is npq without using the Bernoulli distribution and independence way.

Actually not knowing what text material you have been given, this could be seen as an odd question.
Therefore what I do is just a guess. I expect that your text would have shown the if X is the binomial variable then $E\left( {X^2 } \right) = np\left[ {\left( {n - 1} \right)p + 1} \right]$ .
Using the standard definition for variance:
$V(X) = E\left( {X^2 } \right) - E^2 \left( X \right) = np\left[ {\left( {n - 1} \right)p + 1} \right] - \left( {np} \right)^2$ .
From which the result follows.

**eugene1687** · December 6th 2007, 09:48 AM

hey there,
Thank you sooo much

I got the idea on how to show the proof from the help u gave..
Appreciate it..
thanks

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