Statistical Approximation of Streaming Data - Real-Time Analytics - page 389

a special case of the gamma distribution where the shape parameter k is
one-half of the degrees of freedom, and the rate parameter p is one-half.
NOTE
In addition to being the sum of k exponential random variables, a
gamma random variable X ∼ Gamma(a,b) = Y/b, where Y ∼
Gamma(a,1). This property is often used when dealing with Gamma
random variables to simplify the density function.
If two variables X and Y both take on a gamma distribution with the same
parameter for k and different parameters for a “p” then X/(X+Y) takes on a
beta distribution . The beta distribution draws random variables between 0
and 1, and so it is often used to model frequencies.
The uniform distribution on [0,1] is a special case of the beta distribution,
when both a = 1 and k = 1.
Joint Distributions
So far, all of the discussion has focused on a single distribution. This is
useful for investigating a single variable, but most problems are interested
in understanding the relationship between two or more random variables.
In statistics, this relationship between variables is defined by a joint
distribution .
A joint distribution consists of two or more random variables and has a
probability density function and a cumulative density function just like any
other distribution. There is no restriction that the variables are of the same
“type,” and it is common to have distributions that are composed of both
discrete and continuous components.
If two variables are independent, their joint distribution is trivial. The
density functions are simply multiplied. If one or more variables are
conditionally dependent on another variable, the same mechanism used
when defining conditional probability is used to construct the probability
density function of the joint distribution.
Forexample,considerapopulationwherearandomchosenpersonisfemale
with probability p and each subpopulation of male and female having