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5. Functions of a Random Variable
Let X be a r.v defined on the model and suppose g(x) is a function of the variable x. Define
Is Y necessarily a r.v? If so what is its PDF pdf
Clearly if Y is a r.v, then for every Borel set B, the set of for which must belong to F. Given that X is a r.v, this is assured if is also a Borel set, i.e., if g(x) is a Borel function. In that case if X is a r.v, so is Y, and for every Borel set B
(5-1)
(5-2)
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In particular
Thus the distribution function as well of the density function of Y can be determined in terms of that of X. To obtain the distribution function of Y, we must determine the Borel set on the x-axis such that for every given y, and the probability of that set. At this point, we shall consider some of the following functions to illustrate the technical details.
(5-3)
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Example 5.1: Solution: Suppose
and
On the other hand if then
and hence
(5-4)
(5-5)
(5-6)
(5-7)
(5-8)
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From (5-6) and (5-8), we obtain (for all a)
Example 5.2:
If then the event and hence
For from Fig. 5.1, the event is equivalent to
(5-9)
(5-10)
(5-11)
(5-12)
Fig. 5.1
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Hence
By direct differentiation, we get
If represents an even function, then (5-14) reduces to
In particular if ~ so that
(5-14)
(5-15)
(5-16)
(5-13)
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and substituting this into (5-14) or (5-15), we obtain the p.d.f of to be
On comparing this with (3-36), we notice that (5-17) represents a Chi-square r.v with n = 1, since Thus, if X is a Gaussian r.v with then represents a Chi-square r.v with one degree of freedom (n = 1).
Example 5.3: Let
(5-17)
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In this case
For we have and so that
Similarly if and so that
Thus
(5-18)
(5-19)
(5-20)
(5-21)
(a)
(b)
(c)
Fig. 5.2
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Example 5.4: Half-wave rectifier
In this case
and for since
Thus
(5-22)
Fig. 5.3
(5-23)
(5-24)
(5-25)
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Note: As a general approach, given first sketch the graph and determine the range space of y. Suppose is the range space of Then clearly for and for so that can be nonzero only in Next, determine whether there are discontinuities in the range space of y. If so evaluate at these discontinuities. In the continuous region of y, use the basic approach
and determine appropriate events in terms of the r.v X for every y. Finally, we must have for and obtain
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for where is of continuous type.
However, if is a continuous function, it is easy to establish a direct procedure to obtain A continuos function g(x) with nonzero at all but a finite number of points, has only a finite number of maxima and minima, and it eventually becomes monotonic as Consider a specific y on the y-axis, and a positive increment as shown in Fig. 5.4
Fig. 5.4
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Using (3-28) we can write
But the event can be expressed in terms of as well. To see this, referring back to Fig. 5.4, we notice that the equation has three solutions (for the specific y chosen there). As a result when the r.v X could be in any one of the three mutually exclusive intervals
Hence the probability of the event in (5-26) is the sum of the probability of the above three events, i.e.,
(5-26)
(5-27)
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For small making use of the approximation in (5-26), we get
In this case, and so that (5-28) can be rewritten as
and as (5-29) can be expressed as
The summation index i in (5-30) depends on y, and for every y the equation must be solved to obtain the total number of solutions at every y, and the actual solutions all in terms of y.
(5-28)
(5-29)
(5-30)
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For example, if then for all and represent the two solutions for each y. Notice that the solutions are all in terms of y so that the right side of (5-30) is only a function of y. Referring back to the example (Example 5.2) here for each there are two solutions given by and ( for ). Moreover
and using (5-30) we get
which agrees with (5-14).
(5-31)
Fig. 5.5
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Example 5.5: Find
Solution: Here for every y, is the only solution, and
and substituting this into (5-30), we obtain
In particular, suppose X is a Cauchy r.v as in (3-39) with parameter so that
In that case from (5-33), has the p.d.f
(5-33)
(5-32)
(5-34)
(5-35)
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But (5-35) represents the p.d.f of a Cauchy r.v with parameter Thus if ~ then ~
Example 5.6: Suppose and Determine Solution: Since X has zero probability of falling outside the interval has zero probability of falling outside the interval Clearly outside this interval. For any from Fig.5.6(b), the equation has an infinite number of solutions where is the principal solution. Moreover, using the symmetry we also get etc. Further,
so that
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Using this in (5-30), we obtain for
But from Fig. 5.6(a), in this case (Except for and the rest are all zeros).
(5-36)
(a)
(b)
Fig. 5.6
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Thus (Fig. 5.7)
Example 5.7: Let where ~ Determine Solution: As x moves from y moves from From Fig.5.8(b), the function is one-to-one for For any y, is the principal solution. Further
(5-37)
Fig. 5.7
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so that using (5-30)
which represents a Cauchy density function with parameter equal to unity (Fig. 5.9).
(5-38)
(a)
(b)
Fig. 5.8
Fig. 5.9
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Functions of a discrete-type r.v
Suppose X is a discrete-type r.v with
and Clearly Y is also of discrete-type, and when and for those
Example 5.8: Suppose ~ so that
Define Find the p.m.f of Y. Solution: X takes the values so that Y only takes the value and
(5-39)
(5-40)
(5-41)
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so that for
(5-42)
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