Frequently, we observe a value of some random variable, but are really interested in a value derived from this by a function rule. If X is a random variable and g is a reasonable function (technically, a Borel function), then Z=g(X)is a new random variable which has the value g(t) for any ω such that X(ω)=t. Thus Z(ω)=g(X(ω)). Suppose we have the distribution for X. How can we determine P(Z∈M), the probability Z takes a value in the set M?
Mapping approach. Simply find the amount of probability mass mapped into the set M by the random variable X. In the absolutely continuous case, integrate the density function for X over the set M. In the discrete case, as an alternative, select those possible values for X which are in the set M and add their probabilities.For a Borel function g and set M, determine the set N of all those t which are mapped into M, then determine the probability X is in N as in the previous case.
Introduction
Frequently, we observe a value of some random variable, but are really interested
in a value derived from this by a function rule. If
X is a random variable and
g is
a reasonable function (technically, a
Borel function ), then
$Z=g\left(X\right)$ is a new random
variable which has the value
$g\left(t\right)$ for any
ω such that
$X\left(\omega \right)=t$ .
Thus
$Z\left(\omega \right)=g\left(X\right(\omega \left)\right)$ .
The problem; an approach
We consider, first, functions of a single random variable. A wide variety of functions
are utilized in practice.
A quality control problem
In a quality control check on a production line for ball bearings it may be easier to weigh the
balls than measure the diameters. If we can assume true spherical shape and
w is the weight, then diameter is
$k{w}^{1/3}$ , where
k is a factor depending upon the
formula for the volume of a sphere, the units of measurement, and the density of the steel.Thus, if
X is the weight of the sampled ball, the desired random variable is
$D=k{X}^{1/3}$ .
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Price breaks
The cultural committee of a student organization has arranged a special deal
for tickets to a concert. The agreement is that the organization will purchase ten ticketsat $20 each (regardless of the number of individual buyers). Additional tickets are
available according to the following schedule:
 1120, $18 each
 2130, $16 each
 3150, $15 each
 51100, $13 each
If the number of purchasers is a random variable
X , the total cost (in dollars) is
a random quantity
$Z=g\left(X\right)$ described by
$$g\left(X\right)=200+18{I}_{M1}\left(X\right)(X10)+(1618){I}_{M2}\left(X\right)(X20)$$
$$+(1516){I}_{M3}\left(X\right)(X30)+(1315){I}_{M4}\left(X\right)(X50)$$
$$\text{where}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}M1=[10,\phantom{\rule{0.166667em}{0ex}}\infty ),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}M2=[20,\phantom{\rule{0.166667em}{0ex}}\infty ),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}M3=[30,\phantom{\rule{0.166667em}{0ex}}\infty ),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}M4=[50,\phantom{\rule{0.166667em}{0ex}}\infty )$$
The function rule is more complicated than in
[link] , but the essential problem is the same.
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The problem
If
X is a random variable, then
$Z=g\left(X\right)$ is a new random variable.
Suppose we have the distribution for
X . How can we determine
$P(Z\in M)$ , the probability
Z takes a value in the set
M ?
An approach to a solution
We consider two equivalent approaches
 To find
$P(X\in M)$ .

Mapping approach . Simply find the amount of probability mass mapped into
the set
M by the random variable
X .
 In the absolutely continuous case, calculate
$\int}_{M}{f}_{X$ .
 In the discrete case, identify those values
t
_{i} of
X which are in the set
M and add
the associated probabilities.

Discrete alternative . Consider each value
t
_{i} of
X . Select those
which meet the defining conditions for
M and add the associated probabilities. This
is the approach we use in the MATLAB calculations. Note that it isnot necessary to describe geometrically the set
M ; merely use the defining conditions.
 To find
$P\left(g\right(X)\in M)$ .

Mapping approach . Determine the set
N of all those
t which are mapped into
M by the function
g . Now if
$X\left(\omega \right)\in N$ , then
$g\left(X\right(\omega \left)\right)\in M$ , and if
$g\left(X\right(\omega \left)\right)\in M$ , then
$X\left(\omega \right)\in N$ . Hence
$$\{\omega :g(X\left(\omega \right))\in M\}=\{\omega :X(\omega )\in N\}$$
Since these are the same event, they must have the same probability. Once
N is identified,
determine
$P(X\in N)$ in the usual manner (see part a, above).

Discrete alternative . For each possible value
t
_{i} of
X , determine
whether
$g\left({t}_{i}\right)$ meets the defining condition for
M . Select those
t
_{i} which do and
add the associated probabilities.