The geometric distribution can be used to model the number of trials before a certain event occurs. For example, we might flip a coin repeatedly until the first head appears. If the coin is fair, the probability of getting a head on each flip is 0. 5. Furthermore, we may realistically assume that the trials are independent. The flip on which the first head occurs can be represented by a random variable, X.

For the general geometric distribution, we maintain the assumption of independent trails—which, admittedly, is sometimes too strong—but allow the probability of the event occurring on any trial to be θ for any 0 < θ < 1. We assume that this probability is the same from trial to trial. In the coin-flipping example, allowing the coin to be biased toward, say, heads, would mean θ > 0.5. Another example would be an unemployed worker repeatedly interviewing for jobs until the first job offer. Then u is the probability of receiving an offer during any particular interview. To follow the geometric distribution, we would assume u is the same for all interviews and that the outcomes are independent across interviews. Both assumptions may be too strong.

One way to characterize the geometric distribution is to define a sequence of Bernoulli (binary) variables, say W₁, W₂, W₃, . . . . If W_k = 1 then the event occurs on trial k; if W_k = 0, it does not occur. Assume that the Wk are independent across k with the Bernoulli(u) distribution, so that P(W_k = 1) = θ.

(i) Let X denote the trial upon which the first event occurs. The possible values of X are {1, 2, 3, . . .}. Show that for any positive integer k,

P(X = k) = (1 – θ)^k–1 θ.

(ii) Use the formula for a geometric sum to show that

(iii) Suppose you have observed 29 failures in a row. If θ = 0.04, what is the probability of observing a success on the 30th trial?

(iv) In the setup of part (iii), before conducting any of the trials, what is the probability that the first success occurs before the 30th trial?

(v) Reconcile your answers from parts (iii) and (iv).

Transcribed Image Text:

P(X < k) = 1- (1 – 0)*, k = 1, 2, ....