Geometric Distribution

The geometric distribution with parameter p (0 < p < 1) is the discrete probability distribution with the following probability mass function:

p(x;p) = p(1 - p)^x-1 for x = 1, 2, 3, .....

The above function describes a probability distribution because

Consider a statistical experiment where a success occurs with probability p and a failure occurs with probability q = 1 - p. If the experiment is repeated indefinitely and the trials are independent of each other, then the random variable X, whose value is the number of the trial on which the first success occurs, has a geometric distribution and is said to be geometrically distributed. For the first success to occur on the xth trial, the first (x-1) trials must result in a failure; this event happens with probability (1 - p)^x-1. The success on the xth trial will occur with probability p. Hence, the probability of the first success occurring on the first trial is p(1 - p)^x-1.

The mean and variance of a random variable X that is geometrically distributed with parameter p are

EX. A particularly biased coin, when tossed, will come up heads 75 % of the time. The random variable X, whose value is the number of the first toss that results in heads, is geometrically distributed with p = 0.75. Its probability mass function is given by

p(x; 0.75) = 0.75(1 - 0.75)^x-1

= 0.75(0.25)^x-1 for x = 1, 2, 3, ....

The probabilities that the first outcome of heads occurs during the first three trials are calculated as follows:

p(1; 0.75) = 0.75(0.25)^1-1 = 0.75
p(2; 0.75) = 0.75(0.25)^2-1 = 0.75(0.25) = 0.1875
p(3; 0.75) = 0.75(0.25)^3-1 = 0.75(0.25)² = 0.046875

The expected number of tosses needed before a head occurs is

With this biased coin, it only takes 1.333 ... tosses, on average, before a head occurs, as opposed to 1/0.5 = 2 tosses for a regular coin.

The variance of X is