M4L4 Expectation and Moments of Functions of Random Variable Download

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M4L4
Expectation and Moments of Functions of Random Variable
1. Introduction
Functions of random variable are discussed in previous lectures. In this lecture, properties of
random variable, e.g. expectation, moments and moment-generating function of ‘functions of
random variable’ are discussed in detail.
2. Usefulness of the properties of random variables
The properties of random variables are useful for statistical problems in civil engineering.
Standard procedure can be followed to obtain the moments and expectations from the
probability distribution functions, as described in previous lectures. There are different
methods to obtain pdf of the functions of random variable available, though the procedure
might be complex in few cases. For such instances, information of moments of the derived
random variables is very useful.
3. Expectation of functions of discrete random variables
Expectation of the discrete random variable X is given by,
Therefore, the expectation of the function of a discrete random variable,
is
expressed as,
Expectation of the continuous random variable X is given by,
3.1. Properties of Expectation
The following properties of expectation hold good.
a.
, if ‘ ’ is constant. b.
, if ‘ ’ is constant. c.
for two constants ‘ ’ and ‘ ’. d. The expectation operator and functions of random variables do not commute, i.e. 3.2. Example for discrete random variables
Problem 1. The random variable
has a probability mass function (pmf)
and . Find the mean of the function
for
.
Solution. The mean of the function,
3.3. Example for discrete random variables
Problem 2. Assume that the inter-arrival time,
of a vehicle approaching a toll station of a
bridge has an exponential pdf with parameter
. There are
Thus
toll lines in that toll station.
vehicles can be accommodated at a time. Determine the mean arrival time of
vehicles and the coefficient of variation of this arrival time. Assume that the arrivals are
independent of each other (Kottegoda and Rosso, 2008).
Solution. As the inter-arrival time,
variance of
are
and
The total time for the arrival of
where,
follows an exponential distribution, the mean and the
respectively.
vehicle is denoted as . Thus we get,
is the arrival time of the th vehicle
and
Hence the coefficient of variation is:
Thus the variation of arrival time decreases with increase in toll lines, as ‘ ’ is a positive real
number.
4. Moments of functions of random variables
In general, the
And the
th
th
moment of the function of discrete random variable
moment of the function of continuous random variable
is given by:
is given by:
4.1. Moments of functions of random variables about its mean
The
th
moment of the function of discrete random variable
th
moment of the functions of continuous functions
about its mean is given
by:
The
is expressed as:
4.2. Variance of discrete functions
The variance of discrete function,
is expressed as:
4.3. Variance of continuous functions
The variance of discrete function,
is expressed as:
5. Mean and Variance of Linear Function
Let us consider a linear function as,
, where a and b are constants.
The mean values of
is mathematical expectation of
, i.e.
Similarly, variance of
can be expressed as,
6. Expansion of Functions of Random Variable
The function of random variable,
value,
can be expanded in a Taylor series about the mean
.
where derivatives are evaluated at
.
If the series is truncated at linear terms, then the first-order approximate mean and variance of
are obtained.
The variance of function of random variable,
It should be noted that, if the function,
is approximately linear for the entire range of
value , then above two equations will yield good approximation of exact moments.
Problem 3. The length of two rods will be determined by two measurements with an unbiased
instruments with an unbiased instrument that make random error with mean and standard
deviation
in each measurement. Compute the variance in the estimation of the length
and
by the following methods:
a. The two rods are measured separately
b. The sum and difference of the length of two rods are measured instead of individual
lengths (Ang and Tang, 1975)
Solution. a. Let
and
denote the measurement obtained for the two rods, then
where,
are the errors involved in the measurements.
The variance in the estimation of
is:
Similarly,
b. Suppose
denotes that the measured combined length of the two rods and
denotes
the measured difference between the lengths of the two rods. Then,
Solving these two equations, we get,
and
Assuming that the errors are statically independent, the variance in the estimation of
is
therefore,
Similarly,
From this, we can further find that the second method of measuring the length of the rods is
better, since the variance in the estimation of the true lengths and , are smaller.
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