Suppose that X is continuous random variable with PDF f(x) and CDF F(x). (a) Prove that if f(x) > 0 only on a single (possibly infinite) interval of the real numbers then F(x) is a strictly increasing function of x over that interval. [Hint: Try proof by contradiction]. (b) Under the conditions described in part (a), find and identify the distribution of Y = F(x).
Given pdf
and CDF
. Now the property of pdf is
.
Now
That is
is a strictly increasing function of
over the interval
.
Given the transformation
.
The pdf of
is
The distribution of
uniform in the interval
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