1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Number of trials until first success
Figure P3.4: Geometric distribution (Definition P3.7). Each bar represents the probability that the first successful event occurs after a particular number of trials (from one to infinity). The probability of success in any one trial is (a) p = 0.5, (b)p = 0.1.
The mean of a geometric random variable is given by E[X| = l/p. To get more comfortable working with sums, it is worth deriving this fact. We start with Definition P3.2 giving the mean:
This sum is not one of those listed in Appendix Al, but consider taking the derivative of both sides of A1.20 with respect to a, giving us a! 1 = 1/(1 - a)2 (To see this, it might help to think about writing out the summation as a + a2 + a3 + •••)• If we factor out p from (P3.8) and let a = 1 - p, the sum in (P3.8) can be written as ak~x, which equals p/( 1 - a)2. Plugging in a = 1 - p, the mean equals
In a similar fashion, the variance of the geometric random variable is
The number of courtship displays made by a male before he successfully mates might be described by a geometric random variable. Here, each time a male displays is a Bernoulli trial resulting in a mating ("success") or not ("failure"). The key assumption for this process to be described by a geometric distribution is that the probability that a mating attempt succeeds remains constant over time and is not influenced by ("is independent of") the outcome of previous mating attempts. The geometric distribution might also describe the time until extinction of an endangered population that is censused yearly, if the probability of extinction is constant. Thus, with an annual extinction risk of 10% (p = 0.1), the expected time until extinction is ten years (i.e., lip = 1/0.1 = 10). The variance in this case is pretty large (90 years squared). This means that the actual year in which the population goes extinct is very hard to predict, as suggested by Figure P3.4.
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