4.3 Geometric distribution  (Page 13/14)

a discrete random variable (RV) that arises from the Bernoulli trials; the trials are repeated until the first success. The geometric variable X is defined as the number of trials until the first success. The mean is μ = 1 p and the standard deviation is σ = 1 p ( 1 p − 1 ) . The probability of exactly x failures before the first success is given by the formula: P ( X = x ) = p (1 – p ) x – 1 where one wants to know probability for the number of trials until the first success: the xth trail is the first success. An alternative formulation of the geometric distribution asks the question: what is the probability of x failures until the first success? In this formulation the trial that resulted in the first success is not counted. The formula for this presentation of the geometric is: P ( X = x ) = p ( 1 − p ) x The expected value in this form of the geometric distribution is μ = 1 − p p The easiest way to keep these two forms of the geometric distribution straight is to remember that p is the probability of success and (1−p) is the probability of failure. In the formula the exponents simply count the number of successes and number of failures of the desired outcome of the experiment. Of course the sum of these two numbers must add to the number of trials in the experiment.