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Random Variables
A random variable is a real-valued function which exists within the domain of a defined sample space. A random variable is designated by a capital letter, say Y, and the value of Y, depending on the outcome of the experiment, is denoted by a small letter, say y. The sample space is the range of values that the value of Y can be assigned.
Random variables can be either discrete or continuous. A discrete random variable can assume only a finite number of distinct values, such as zero or one for example. A continuous random variable can assume any value within the range of the sample space.
Discrete Random Variables
In the discrete case, the probability that Y takes the value y,
is defined as the sum of the probabilities of all sample points that
are assigned the value y. That is,
y | p(y) |
0 | 1/8 |
1 | 1/4 |
2 | 1/4 |
3 | 3/8 |
Any other values of y are assumed to have p(y)=0, and the sum of
the probabilities is 1.
The expected value
of a discrete random variable is defined as
The variance of discrete random variable Y is
Binomial Distribution
A common discrete distribution is the binomial distribution.
A binomial event can take on only two possible outcomes, success or
failure, zero or one, heads or tails, diseased or not diseased,
and so on. The probability of one outcome is q and the probability
of the other outcome is 1-q. Trials, or a succession of binomial
events, are assumed to be independent. The random variable Y is
the number of successes. The probability distribution is given by
Poisson Distribution
A Poisson probability distribution provides a good model for the
probability distribution of the number Y of rare events that
occur in a given space, time, volume, or any other dimension,
and
is the average value of Y. An example in animal
breeding might be the number of quality embryos produced by a
cow during superovulation, which can range from 0 to 20 (or more).
The Poisson probability distribution is given by
General Results
Expectations
If Y represents a random variable from some defined population, then
the expectation of Y is denoted by
The mean of a population is also known as the first moment of the distribution. The exact form of the distribution for a random variable will determine the form of the estimator of the mean and of other parameters of the distribution.
Variance-Covariance Matrices
The variance of a scalar random variable, Y, is defined as
A variance-covariance (VCV) matrix of a random vector contains variances on the diagonals and covariances on the off-diagonals. A VCV matrix is square, symmetric and should always be positive definite or positive semi-definite. Another commonly used name for VCV matrix is dispersion matrix.
Let
be a matrix of constants conformable for multiplication
with the vector ,
then
Continuous Distributions
Consider measuring the amount of milk given by a dairy cow at a
particular milking. Even if a machine of perfect accuracy was used,
the amount of milk would be a unique point on a continuum of
possible values, such as 32.35769842.... kg of milk. As such it is
mathematically impossible to assign a nonzero probability to all
of the infinite possible points in the continuum. Thus, a different
method of describing a probability distribution of a continuous
random variable must be used. The sum of the probabilities (if they
could be assigned) through the continuum is still assumed to sum
to 1. The cumulative distribution function of a random
variable is
If F(y) is the cumulative distribution function of Y, then the
probability density function of Y is given by
The Uniform Distribution
The basis for the majority of random number generators is a
uniform distribution. A random variable Y has a continuous
uniform probability distribution on the interval
if and only if the density function of Y is
The Normal Distribution
A random variable Y has a normal probability distribution if and
only if
For the random vector, ,
the multivariate normal density
function is
Chi-Square Distribution.
The t-distribution.
The t-distribution is based on the ratio of two
independent random variables.
The first is from a univariate normal distribution, and the second is
from a central chi-square distribution. Let
and
with x and u being independent, then
The F-distribution.
The central F-distribution is based on the ratio of two independent
central chi-square variables. Let
and
with u and w being independent, then
The square of a t-distribution variable gives a variable that has an F-distribution with 1 and n degrees of freedom.
Noncentral F-distributions exist depending on whether the numerator or denominator variables have noncentral chi-square distributions. Tables for noncentral F-distributions generally do not exist because of the difficulty in predicting the noncentrality parameters. However, using random chi-square generators it is possible to numerically calculate an expected noncentral F value for specific situations. When both the numerator and denominator chi-square variables are from noncentral distributions, then their ratio follows a doubly noncentral F-distribution.
Quadratic and Bilinear Forms
Quadratic Forms
A quadratic form is a sum of squares of elements of a vector. The general form is , where is a vector of random variables, and is a regulator matrix. The regulator matrix can take on various forms and values depending on the situation. Usually is a symmetric matrix. Examples of different matrices are as follows:
The expected value of a quadratic form is
The quadratic form,
,
has a chi-square distribution if
The covariance between two quadratic forms, say
and
,
is
Bilinear Forms
A bilinear form is represented as
,
where
and
are two different random vectors, possibly of different
lengths, and
is the regulator matrix. If
and
with
,
then
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Larry Schaeffer