Interpretation of random variable

[Obraz35]

Homework Statement

The probability mass function of a random variable X is:
P(X=k) = (r+k-1 C r-1)p^r(1-p)^k
Give an interpretation of X.

Homework Equations

The Attempt at a Solution

The PMF looks like the setup for a binomial random variable. The first combination looks like you are arranging r-1 successes in r+k-1 slots. And the p^r seems like it is giving the probability of r successes occurring. But I don't see what X as a whole is standing for.

 

[lanedance]

i think you;re almost there, so i assume you mean
$$P(X=k) = C_{r-1}^{r+k-1}p^r(1-p)^k$$

can re-write this as
$$P(X=k) = p.C_{(r-1)}^{k+(r-1)}p^{r-1}(1-p)^{k}$$

which as you say comparing with the binomial distribution is for n trials, m success, and probability of success of p

can re-write this as
$$P(M=m) = p.C_{m}^{n}p^{n-m}(1-p)^{m}$$

so your distribution is effectively p (the probability of a single success) times the probability of r-1 successes out of r-1+k trials (with probability of success p).

any ideas for an interpretation of this as a total entity?