M4L5
Expectation and Moments of Functions of Random Variable
1.
Introduction
This lecture is a continuation of previous lecture, elaborating expectations, moments and
moment generating functions of the ‘functions of random variable’ discussed earlier.
2.
Moments of functions of random variables
In general, the
th
moment of the function of discrete random variable
is given by:
And the
th
moment of the function of continuous random variable
is given by:
2.1.
Moments of functions of random variables about its mean
The
th
moment of the function of discrete random variable
about its mean is given
by:
The
th
moment of the functions of continuous functions
is expressed as:
2.2.
Variance of discrete functions
The variance of discrete function,
is expressed as:
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2.3.
Variance of continuous functions
The variance of discrete function,
is expressed as:
3.
Mean and Variance of Linear Function
Let us consider a linear function as,
, where
and
are constants.
The mean values of
is mathematical expectation of
, i.e.
Similarly, variance of
can be expressed as,
Problem 1.
The random variable
has a probability mass function (pmf)
for
and
. Find the variance of the function
.
Solution.
The mean of the function,
Problem 2.
Consider a simple case where the variable can only takes a value of
or
. This
situation can represent the occurrence of a flood at a particular site on a river, where the event
is the exceedence of a specified flow in the river. Let the probability of such an occurrence
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 Spring '13
 Dr.RajibMaity
 Civil Engineering, Probability theory, probability density function, Kottegoda

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