Summing up the Series: $\sum^{n}_{x=2}x(x-1)\binom{n}{x}p^{x}q^{n-x}$

[hanboning]

The series in question is \sum^{n}_{x=2}x(x-1)\binom{n}{x}p^{x}q^{n-x}. How do I find the sum of this?

I think this is related to expected values and doing cancelling with the x(x-1) with the combination.

Thanks!

 

[tiny-tim]

welcome to pf!

hi hanboning! welcome to pf!

hint: simplify x(x-1) ⁿC_x

 

[hanboning]

Hi tim, thanks for your help. I might have gotten it.

So I simplified the things you said in your hint and I got n(n-1)\sum^{n}_{x=2}{{n-2}\choose{x-2}}p^{x}q^{n-x}. Then I moved a p^{2} out of the sum and got n(n-1)p^{2} as the answer.

 

[tiny-tim]

s'right!

(i'm assuming p + q = 1? )

 

[hanboning]

Yep, I appreciate your help. Happy new year.