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
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