Many patients drop out of IVF programmes without achieving a successful pregnancy. In this question we’ll see how to use the data from these patients when we calculate maximum likelihood estimates, along with the data from patients who did achieve a successful outcome.

a. Let Z ∼ Geometric(p). Briefly explain why the following formula is true:

P(Z ≥ z) = (1 − p)

z

.

b. A study of IVF success rates followed ten women in their late thirties. Seven of the women achieved a successful pregnancy after respectively 0, 3, 1, 0, 2, 1, and 2 failed cycles. Three of the women dropped out of the programme after respectively 1, 4, and 4 failed cycles. Write out a likelihood function for estimating p that uses the data from all ten women. Hence find the maximum likelihood estimate of p.