# Discrete prob. dist.

1. Oct 28, 2009

### hoffmann

how do i interpret this probability distribution:

$$\sum_{k=r}^\infty \binom{k}{r}p^k(1-p)^{k-r}$$

where r is the number of successes, p is the probability, k trials.

by looking at it, it seems like it's similar to a negative binomial distribution once you pull out a k/r. if you do some math after pulling out the k/r, it seems like it is the expected value of a geometric distribution. is this distribution saying that a negative binomial divided by the number of successes r means there is only one success, which is geometric?

2. Oct 28, 2009

### hoffmann

in other words, what does this sum equal?

3. Oct 29, 2009

### bpet

Do you mean to ask "what is a probabilistic interpretation of this summation formula?" - in which case there would be lots of different ways to write it as an expected value E[f(X)] = sum(f(k)*Prob[X=k]). One way you found was with negative binomial X and f(X)=X/r; the other way was E[g(Y)] for some function g where Y is geometric.

However it sounds like you want to use E[f(X)]=E[g(Y)] to draw conclusions about how X and Y are related, but that's just not possible.

Does that help?

4. Oct 29, 2009

### hoffmann

makes sense. thanks!