Random Variables
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- Statistics
Introduction
It frequently occurs that in performing an experiment, we are mainly
interested in some function of the outcome rather than the outcome
itself. These quantities of interest, are real-valued functions
defined on the sample space
Discrete Random Variables
For a discrete random variable, we define the probability mass
function
The probability mass function
Since
The cumulative distribution function
Bernoulli Random Variable
A random variable
This corresponds to the outcome of a trial with binary outcomes.
Binomial Random Variable
Suppose
The pmf of a binomial random variable is given by:
Geometric Random Variable
Suppose that independent trials, each having probability
Poisson Random Variable
A random variable
An important property of the Poisson random variable is that it may be
used to approximate a binomial random variable when the binomial
parameter
Continuous Random Variables
Continuous random variables have an uncountable set of possible values.
The cumulative distribution
Uniform Random Variable
Exponential Random Variables
Gamma Random Variables
The quantity
and
Normal Random Variables
if
Expectation of Random Variables
The expected value of a discrete random variable
and for a continuous random variable is defined similarly:
Suppose we are interested in getting the expected value of a function
of a random variable
We also have linearity of expectations:
The variance of
Jointly Distributed Random Variables
The joint cumulative probability distribution function of
The distribution of
Covariance and Variance of Sums of Random Variables
The covariance of any 2 random variables
If
Some properties of covariance: