That means that if you intend to repeat an experiment until the first success, then, given that the first success has not yet occurred, the conditional probability distribution of the number of additional trials does not depend on how many failures have been observed. The die one throws or the coin one tosses does not have a "memory" of these failures. The geometric distribution is the only memoryless discrete distribution. Among all discrete probability distributions supported on {1, 2, 3, ... } with given expected value μ, the geometric distribution X with parameter p = 1/μ is the one with the largest entropy. The geometric distribution of the number Y of failures before the first success is infinitely divisible, i.e., for any positive integer n, there exist independent identically distributed random variables Y1, ..., Yn whose sum has the same distribution that Y has. These will not be geometrically distributed unless n = 1; they follow a negative binomial distribution. The decimal digits of the geometrically distributed random variable Y are a sequence of independent (and not identically distributed) random variables. For example, the hundreds digit D has this probability distribution: where q = 1 − p, and similarly for the other digits, and, more generally, similarly for numeral systems with other bases than 10. When the base is 2, this shows that a geometrically distributed random variable can be written as a sum of independent random variables whose probability distributions are indecomposable. Golomb coding is the optimal prefix code for the geometric discrete distribution. The sum of two independent Geo(p) distributed random variables is not a geometric distribution. Related distributions The geometric distribution Y is a special case of the negative binomial distribution, with r = 1. More generally, if Y1, ..., Yr are independent geometrically distributed variables with parameter p, then the sum follows a negative binomial distribution with parameters r and p. The geometric distribution is a special case of discrete compound Poisson distribution. If Y1, ..., Yr are independent geometrically distributed variables (with possibly different success parameters pm), then their minimum is also geometrically distributed, with parameter Suppose 0 < r < 1, and for k = 1, 2, 3, ... the random variable Xk has a Poisson distribution with expected value r k/k. Then has a geometric distribution taking values in the set {0, 1, 2, ...}, with expected value r/(1 − r). The exponential