Geometric Distribution
Geometric Distribution
In probability theory and statistics, the geometric distribution is either of two discrete
probability distributions:
The probability distribution of the number of X Bernoulli trials needed to get one success,
supported on the set { 1, 2, 3, ...}
The probability distribution of the number Y = X - 1 of failures before the first success,
supported on the set { 0, 1, 2, 3, ... }
Which of these one cal s "the" geometric distribution is a matter of convention and
convenience.
These two different geometric distributions should not be confused with each other. Often, the
name shifted geometric distribution is adopted for the former one (distribution of the number
X); however, to avoid ambiguity, it is considered wise to indicate which is intended, by
mentioning the range explicitly.
It's the probability that the first occurrence of success require k number of independent trials,
each with success probability p.
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If the probability of success on each trial is p, then the probability that the kth trial (out of k
trials) is the first success is
for k = 1, 2, 3, ....
The above form of geometric distribution is used for modeling the number of trials until the first
success. By contrast, the following form of geometric distribution is used for modeling number
of failures until the first success:
for k = 0, 1, 2, 3, ....
In either case, the sequence of probabilities is a geometric sequence. For example, suppose
an ordinary die is thrown repeatedly until the first time a "1" appears.
The probability distribution of the number of times it is thrown is supported on the infinite set
{ 1, 2, 3, ... } and is a geometric distribution with p = 1/6.
Like its continuous analogue (the exponential distribution), the geometric distribution is
memoryless.
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 in fact the only memoryless discrete distribution.
Among all discrete probability distributions supported on {1, 2, 3, ... } with given expected
value , the geometric distribution X with parameter p = 1/ 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.
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These wil not be geometrically distributed unless n = 1; they fol ow a negative binomial
distribution.
The decimal digits of the geometrical y distributed random variable Y are a sequence of
independent (and not identical y distributed) random variables. For example, the hundreds
digit D has this probability distribution:


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