A discrete (infinite) random variable is a random
variable which may take a discrete though infinite set of
possible values. For the sake of simplification, we
assume that the possible values are the non-negative integers.
We present such a random variable by giving a sequence
of relative proportions. As with
finite random variables, we presume that
for all
and that
.
We regard as the probability that
takes the value
.
If is a discrete random variable with
, then
and
The Poisson distribution with parameter is the discrete
probability distribution with
.
If has this distribution, then
A similar calculation shows that
.
A certain website has an average of twenty hits per hour and the number of such hits is Poisson distributed. What is the probability that the website has ten or fewer hits in some hour?
Fix a number with
. A discrete infinite random variable
is a geometric random variable with success probability
if
the relative frequency of
is
.
Using a geometric series one may compute that
while
a Taylor series computation shows that
.
If one has a sequence of independent experiments where the probability of
``success'' being , then the random variable which expresses the probability of
failures before the first success is geometrically distributed with success
probability
.
If one has a coin which comes up tails with a eighty percent probability, what is the probability that the first tails appears after a string of four heads?
We may model this situation with a geometric random variable for which ``success''
means obtaining heads. So, the success probability is . The probability
that the first tails is obtained after a string of four heads is then
.