Lecture 15/part2 Limiting Distribution/2017 2018 /

Transformation or One-dimensional Random Variable.
Let X be a random vanable defined on abe event space S and let g (.) be a function such that Y = g (X) is also defined on the event space S. In this section we shall deal with the following problem :
"Given the probability density of a r.v. X, to determine the density of a new r.v.Y=•g(X)."
It can be proved in general that.. if g (.) is any continuous function, then the distribution of Y = g ( X) is uniquely determined by that of X . The proof of this result is rather difficult and beyond the scope of this book. Here we shall consider
the following, relatively simple theorem.
Theorem 5•9. Let X be a continuoUs r.v. withp.df.fx (x). Let y= g(x)
be strictly monotonic (increasing or decreasing) function of x. Assume that g (x) is differentiable (and hence continuous)for aUx. Then the p.d.f. oft he r.v.
Y is given by
h(y)= f(x) |J|
where x is expressed in terms of y.
Note that strictly speaking the probability density function fx( ) of a random variable X is not uniquely defined. All that the definition requires is that the integral of fx( ) gives Fx(x) for every x, and more than one function fx( ) may satisfy such requirement.
considerations and for this reason