Download M5L7 Moment Generating Function of Multivariate RVs and

M5L7
Moment Generating Function of Multivariate RVs and Multivariate
Probability Distributions
1. Introduction
In this lecture, moment generating functions, bivariate probability distribution function and
joint probability of multivariate random variables are discussed in details. Numerical
problems are also solved to explain the theories.
2. Joint Moment Generating Function
Similar to moment-generating function (MGF) of a random variable defined in previous
module, the moment generating function for two random variables is defined for discrete and
continuous cases separately.
2.1. MGF for Discrete case
The moment generating function for two discrete random variables can be obtained as,
2.2. MGF for Continuous case
The moment generating function for two discrete random variables can be obtained as,
2.3. MGF for Marginal
The moment generating function for marginal can be derived as,
2.4. MGF for Statistically Independent RVs
If
and
and are two statistically independent RVs, their joint MGF can be obtained as,
2.5. MGF for Multiple RVs
The joint moment-generating function of multivariate random variables
also defined similar to two RVs as,
The
moment of
function
can
can be determined by differentiating the joint moment generating
times with respect to
and then evaluating derivative with
.
where
The mixed moments of
differentiating
derivative with
and
,
times with respect to
.
can be generated from the joint mgf by
and
times with respect to
. Then evaluating the
Problem 1. Life of a structure consist of two subcomponents, depends on their individual life
time
and
is given by,
, which are exponentially distributed. The joint density function of
Determine the joint moment generating function of
and
. Compute the mean of
and
and
. Also compute the covariance between them.
Solution. We have,
So we get,
Then to find
,
Similarly to find
To get
,
,
And to get
,
3. Bivariate Probability Distribution
There various probability distribution functions are applicable to multiple RVs, similar to
distribution of simple RVs. Here exponential and normal probability distribution functions
for bivariate RVs are discussed.
3.1. Bivariate Exponential Distribution
The pdf and cdf of the Bivariate Exponential Distribution can be defined as,
for
;
and
3.2. Bivariate Normal Distribution
Bivariate Normal Distribution is an example of joint density function of two continuous RVs,
say,
and
. In such cases, marginal density functions of both the RVs are also Normal
distribution. However, reverse is not true, i.e. if both
and
are normally distributed their
joint distribution not necessarily be a joint Normal distribution.
The joint pdf of bivariate normal distribution is expressed as,
for
where,
and
is the mean of
;
is the mean of
is the correlation coefficient between
Now taking
Distribution is expressed as,
;
and
and
is the variance of
;
variance of
.
, the bivariate standard Normal
for
The volume enclosed by the surface of bivariate normal pdf (as shown in fig. 1) is unity.
y
2
y
x
1
1
x
2
Fig. 1. Surface of Joint Normal Distribution of
and
The Bivariate Normal Density function equation can be expanded as,
Now the conditional density function of
And, marginal density function of
given
can be obtained as,
is,
Also the conditional density function has a mean as given,
and the variance is given as,
4. Probability Distribution for Multivariate RVs
So far discussion on probability distribution for RVs has been concentrated on two RVs and
their distribution functions. In many real cases, there would more than two random variables
representing different aspects of the same experiment and same sample space. Henceforth,
the concepts of probability distribution will be extended for multivariate RVs having more
than two variables.
4.1. n-dimensional Discrete Random Variable
If
are discrete random variables defined on the same probability space, then
is defined as a n-dimensional discrete random variable if it can take values
. These variables
are said
only at a countable number of points
to be n-dimensional discrete random variables.
4.1.1. Joint Probability Mass Function (Joint pmf)
The joint pmf of an
-dimensional random variable
intersection probability of the n-sequence of events
are points in the
is defined as the
if
-dimensional sample space of this variable, and 0 for other
cases.
The properties of joint pmf are similar to pmf as given,
1.
p X 1 ,..., X n  x1 ,..., xn   0
2.
p
X 1 ,..., X n
x1 ,..., xn   1
X 1 ,..., X n
4.1.2. Joint Cumulative Distribution Function (Joint cdf)
The joint cumulative distribution function for n-dimensional discrete random variables is
given by,
4.2. n-dimensional Continuous Random Variable
If
are -dimensional random variables defined on the same probability space,
is defined as an n-dimensional continuous random variable if and only if
then
there exists a function such that
and
4.2.1. Joint Probability Distribution Function (Joint pdf)
The joint pdf of an
-dimensional random variable
intersection probability of the random variables and expressed as,
is defined as the
4.2.2. Joint Cumulative Distribution Function (Joint cdf)
The joint distribution of
their joint cdf as,
continuous random variable can be completely described with
Problem 1. The joint density of random variables is given by:
Calculate the probability
Solution. The joint pdf can be used to compute probabilities as: