Binomial random variable help

#1
If X = a binomial random variable with parameters n and p, with 0<p<1

How do I show that P[X=k+1] = [p/(1-p)][(n-k)/k+1)]P[X=k]

Also, how do I prove that as k goes from 0 to n, P[X=k] first increases and then decreases, with it reaching largest value when k is the largest integer less than or equal to (n+1)p.

Thanks!

I have no idea how to even start this problem
 

BioStatMatt

TS Contributor
#2
Try this:

P[X = k+1] = n!/(k+1)!(n-k-1)! p^(k+1) (1-p)^(n-k-1)
= n!(n-k)/[k+1(k!)(n-k)!] pp^(k) (1-p)^(-1) (1-p)^(n-k)
= (n-k)/(k+1) p/(1-p) n!/[k!(n-k)!] p^(k) (1-p)^(n-k)

Now, we know that the last 3 terms of the above equality defines the pmf of a binomial random variable, which proves the first part of your problem.

For the second part, remember that the pmf evaluated at each point in the support must sum to 1. You will find that only when k goes from 0 to n will this happen.

~Matt