True/False — Geometric and Negative Binomial concepts.
1) In the geometric distribution (trials-to-first-success version), X can take values 1,2,3,…
2) For geometric(p) with X = trial of first success, the CMF is P(X=k) = (1−p)k−1 p for k ≥ 1.
3) The geometric distribution is NOT memoryless: P(X > m+n | X > m) = P(X > n).
4) For geometric(p) (trials-to-first-success), the mean is 1/p and the variance is (1−p)/p2.
5) Negative binomial(r,p) models the number of successes in a fixed number of trials.
6) In the “trials-to-r-th-success” parameterization, NB(r,p) has PMF P(X=k) = C(k−1, r−1) pr(1−p)k−r, for k = r, r+1, …
7) For NB(r,p) (trials-to-r-th success), the mean is r/p and the variance is r(1−p)/p2.
8) Geometric(p) is a special case of binomial with r=1.
9) If p increases, the expected waiting time in geometric(p) (trials to first success) increases.
10) Negative binomial is appropriate when you count failures before the r-th success or trials up to the r-th success (with the right parameterization), not when n is fixed beforehand.