Fundamentals of Probability and Statistics for Engineers

where g(X) is assumed to be a continuous function of X. Given the probability
distribution of X in terms of its probability distribution function (PDF),
probability mass function (pmf ) or probability density function (pdf ), we
are interested in the corresponding distribution for Y and its moment
properties.

5.1.1 Probability D istribution

Given the probability distribution of X, the quantity Y, being a function of X as
defined by Equation (5.2), is thus also a random variable. Let RX be the range
space associated with random variable X, defined as the set of all possible
values assumed by X, and let RY be the corresponding range space associated
with Y. A basic procedure of determining the probability distribution of Y
consists of the steps developed below.
For any outcome such as X x, it follows from Equation (5.2) that
Y y g(x). As shown schematically in Figure 5.1, Equation (5.2) defines a
mapping of values in range space RX into corresponding values in range space
RY. Probabilities associated with each point (in the case of discrete ra ndom
variable X) or with each region (in the case of continuous random variable X) in
RX are carried over to the corresponding point or region in RY. The probability
distribution of Y is determined on completing this transfer process for every
point or every region of nonzero probability in RX. Note that many-to-one
transformations are possible, as also shown in Figure 5.1. The procedure of
determining the probability distribution of Y is thus critically de pe ndent on the
fu nctional form of g in Equation (5.2).

X=x 3

X=x 2

X=x 1

X=x

RY

RX

Y=y=g(x 1 )=g(x 2 )=g(x 3 )

Y=y=g(x)

Figure 5. 1 Transformation y g(x)

120 Fundamentals of Probability and Statistics for Engineers

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