Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 805 — #813
19.3. Distribution Functions 805
Both PDF R and CDF R capture the same information about R, so take your choice.
The key point here is that neither the probability density function nor the cumulative
distribution function involves the sample space of an experiment.
One of the really interesting things about density functions and distribution functions
is that many random variables turn out to have the same pdf and cdf. In other
words, even though R and S are different random variables on different probability
spaces, it is often the case that
PDF R D PDF S :
In fact, some pdf’s are so common that they are given special names. For example,
the three most important distributions in computer science are the Bernoulli
distribution, the uniform distribution, and the binomial distribution. We look more
closely at these common distributions in the next several sections.
19.3.1 Bernoulli Distributions
A Bernoulli distribution is the distribution function for a Bernoulli variable. Specifically,
the Bernoulli distribution has a probability density function of the form
f p W f0; 1g ! Œ0; 1 where
f p .0/ D p;
f p .1/ D 1 p;
for some p 2 Œ0; 1. The corresponding cumulative distribution function is F p W
R ! Œ0; 1 where
8
ˆ< 0 if x < 0
F p .x/ WWD p if 0 x < 1
ˆ:
1 if 1 x:
19.3.2 Uniform Distributions
A random variable that takes on each possible value in its codomain with the same
probability is said to be uniform. If the codomain V has n elements, then the
uniform distribution has a pdf of the form
where
for all v 2 V .
and
f W V ! Œ0; 1
f .v/ D 1 n