8
Functions of Random
Variables
Learning Objectives
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The purpose of this section is to:
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Describe the problem of finding the distribution of a function of
random variables
This modeling is required for propagation of uncertainties in risk
and reliability analysis
Describe the basic analytical and approximate methods for
estimating the mean and variance of a function of random
variables
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They can be used to approximate the distribution of the function
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Section 8. Functions of Random Variables
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UN 0701 Engineering Risk and Reliability
Section 8. Functions of Random Variables
8.1 Introduction
Fundamental Problem
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Many engineering problems involve a functional relationship
between a system response variable and one or more basic
(independent) variables
If any of the basic variables are random, the response variable
will also be random
The probability distribution and the moments of the dependent
variable will also be functionally related to the basic random
variables
How do we estimate the probability distribution of the
dependent variable?
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How can we estimate mean and standard deviation of the
function
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Section 8. Functions of Random Variables
8.2 Distributions of Functions of RVs
Functions of RVs
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Types of functions encountered in risk and reliability problems
can be classified as
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Scalar Function – representing a single response quantity
Vector Function – representing multiple responses
Scala Functions
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Univariate function: depends on a single RV,
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Multivariate function: depends on multiple RVs,
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(1) Linear, or (2) Nonlinear functions
(1) Linear, or (2) Nonlinear functions
Vector Functions
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More complex and not covered in this course
Analysis of Functions of RVs
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The analysis of a “random” function means
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Probability distribution of a function
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Determine the complete probability distribution
Determine the moments (mean and SD) only
Transformation method based on calculus concepts
Easy to apply to univariate (scalar) and linear functions
For multivariate functions, only simple cases can be solved
Moments of a function
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Since distribution of a function is analytically complex, it is easier
to derive moments of the function
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Exact solutions are possible for simple functions (linear functions)
For complex functions, approximations are based on the Taylor series
Simulation Method: A general numerical approach to solve
these problems
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Section 8. Functions of Random Variables
Functions of Single RVs
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The probability distributions of functions of random variables
can be derived through transformation
Consider the function of a single random variable
where X is a random variable and g(x) is a monotonically
increasing function of x
Generally we can solve y = g(x) for x
y
to find the inverse function
e.g. y = ex
where g-1 is the inverse function of g
x
Distribution of the Function
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Under these conditions, we can solve directly for the CDF of
the dependent variable Y as
or
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In general when finding the distributions of functionally related
random variables, we must work with their CDF’s
To obtain the PDF of Y, we must find the CDF and then
differentiate it, therefore, given
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Section 8. Functions of Random Variables
PDF of Y
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The result is
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Replacing g-1(y) with x we obtain
or
for functions where y increases with x
Graphical Interpretation
y
y = g(x)
dy
dx
x
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The likelihood that Y takes on a value in an interval of width dy
centered on the value of y is equal to the likelihood that X takes on
a value in an interval centered on the corresponding value
x = g-1(y), but with a width dx = dg-1(y)
The interval widths are generally not equal because of the slope
of the function g(x) at y
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Section 8. Functions of Random Variables
Monotonically Decreasing Functions
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For monotonically decreasing functions, we have
which results in
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However, because the first term on the right hand side is
negative we get
or
where |..| denotes the absolute value
Example 8.1
Consider a linear relationship between X and Y as Y = aX + b.
Find the distribution of Y given that X N(X,X).
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Solution: The inverse function is
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Recall the PDF for the Normal distribution, we get
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Therefore, Y follows the Normal distribution with Y = aX + b
and Y = aX
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Section 8. Functions of Random Variables
Example 8.2
If X has a LOG-NORMAL distribution with parameters l and z,
what is the distribution of Y = ln(X)?
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Solution: The inverse function is
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Recall the PDF for the Log-Normal distribution, we get
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Therefore, Y is a Normal variate with the mean value l and
standard deviation z. That is, E[ln(X)] = l and Var[ln(X)] = z2
Functions of Two RVs
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Consider, for example, the following functional relationship
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We want to map f(x,y) into g(z,y) or g(x,z)
The joint probability distribution function (CDF) of X and Y is
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and the joint PDF is
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The CDF of Z is given as
UN 0701 Engineering Risk and Reliability
Section 8. Functions of Random Variables
Transformation
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Consider the mapping of f(x,y) to g(z,y)
The inverse function is equal to x = z - y
The transformation is
where the Jacobian J(z,y) is equal to
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Therefore,
Joint PDF of Z
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The joint PDF is obtained by taking the derivative of the above
distribution function as
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For independent X and Y
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Therefore, the joint PDF becomes
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Section 8. Functions of Random Variables
Joint PDF of Z (cont’d)
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It can also be shown that for the mapping of f(x,y) to g(x,z)
and for independent X and Y
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The above equations are convolution integrals
In many problems, performing the integration may be
challenging, especially since many practical PDF’s are defined
by different functions (e.g. Weibull, Gumbel, etc.)
Need to use numerical methods such as Monte Carlo
Simulation, or approximation methods such as FORM and
SORM
Example 8.3
Consider the special case where X and Y are statistically
independent Normal random variables with means and standard
deviations X, X and Y, Y, respectively. Find the probability
density for Z = X – Y.
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Solution: The probability density is
UN 0701 Engineering Risk and Reliability
Section 8. Functions of Random Variables
Solution (cont’d)
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Simplifying the power terms
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Substituting back
where
and
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After integration and algebraic reduction..., the final result for
the density function of Z becomes
which is a Normal density function with mean and standard
deviation of
and
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In conclusion, any linear function of Normal random variables
is also a Normal random variable
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Section 8. Functions of Random Variables
8.3 Moments of Functions of RVs
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In many cases, the moments (e.g. mean and variance) of the
function may be obtained more easily than the full distribution
of the function
The moments may be sufficient for practical purposes even if
the correct probability distribution cannot be determined
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With knowledge of mean and standard deviation, the COV can be
estimated, which is a relative measure of variability
This is meant as “propagation of uncertainty”
A function’s distribution can be approximated by a Normal
distribution, if its mean and SD are known
Similar to the probability distributions, the moments of functions
of random variables are related to the moments of the basic
random variables
Mean and Variance of Linear Functions
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Consider a univariate linear function Y = aX + b where a and b
are constants
The mean value of Y is given by the mathematical expectation
of the function as
The variance of Y is given as
UN 0701 Engineering Risk and Reliability
Section 8. Functions of Random Variables
Linear Functions (cont’d)
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Consider a bivariate linear function
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where a1 and a2 are constants
The mean value of Y is then
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Therefore
The expected value of a sum is the
sum of the expected values.
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The corresponding variance is given as
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where Cov[X1,X2] is the covariance between X1 and X2
If X1 and X2 were statistically independent, then Cov[X1,X2]
would be equal to zero and the variance would reduce to
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Section 8. Functions of Random Variables
Linear Functions (cont’d)
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For a linear multivariate function
where ai are constants the mean is equal to
and the variance would be
where rXi,Xj is the correlation coefficient between Xi and Xj
Mean and Variance of a Product
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Consider the product of n independent random variables as
(product is a non-linear function)
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The mean value is
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Therefore,
It can also be shown that
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Therefore, the variance of the independent random variables is
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UN 0701 Engineering Risk and Reliability
Section 8. Functions of Random Variables
Function of a Single RV
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Consider a general function (can be non-linear) of a single
random variable X
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The exact moments of Y are given by the mathematical
expectation of g(X)
The mean would be
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and the variance
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Note that the density function fX(x) may not always be known,
and even in cases when it is known, the integrations may be
difficult to perform
Taylor Series Expansion
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To avoid integration, we can approximate the function using
Taylor series expansion
The Taylor series expansion of the function g(x) about a point
x = a is
where g(n)(a) is the nth derivative of g with respect to x at point a
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Section 8. Functions of Random Variables
Approximation of Mean
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Substituting the first 3 terms of the Taylor series expanded at
the mean point for the single variable case
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Assuming the second-order term is negligible, the first-order
approximation of the mean is
Approximation of Variance
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Similarly for the variance, substituting the first 3 terms of the
Taylor series expanded at the mean point
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Ignoring the higher order moments
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Section 8. Functions of Random Variables
General Function of Multiple RVs
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Consider a general function of n random variables
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The mean of Y is given by the mathematical expectation
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The variance is
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Similar to the single variable case, the mean and variance can
be approximated using Taylor series expansion
Approximation of Mean
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Expanding the general function about the mean values of Xn
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Ignoring the higher order terms gives the first-order
approximation of the mean
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Section 8. Functions of Random Variables
Approximation of Variance
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The first-order approximation of the variance is
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If the random variables are uncorrelated
Summary
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Functions are often used to describe the overall response of a
system that may depend on other random variables
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The uncertainty associated with a function can be described by
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e.g. PRA for core damage frequency analysis
Full probability distribution
Mean, SD and COV
This problem can be solved for analytically simple form of
functions (e.g., linear or product function)
In case of complex functions, approximations are required
Simulation is a versatile, numerical method that is universally
used to solve this class of problems
UN 0701 Engineering Risk and Reliability