- Uniform (0, 1)-distribution and its related distributions
Let
be a Uniform(0,1) random variable.
(a) Find the cumulative distribution function of
and its expectation.
(b) Show that
is
-distributed.
(c) Let
be the cumulative distribution function of a continuous random variable. Find the cumulative distribution function of
.
Dieses Beispiel ist als
solved markiert. Ist dies falsch oder ungenau? Aktualisiere den Lösungsstatus (Details:
Vorlage:Beispiel)
--Lessi 2024-02-07T13:04:11Z
Let
be a Uniform(0, 1) random variable, e.g.
Let
be a random variable. To determine its cdf we first define the pdf and cdf of X:
Now we can determine the cdf of
:
To get the expected value we first determine the pdf of
by taking
the derivative of
:
So finally the
expected value is computed
Let
be a random variable. Is
? We want to show that
So again we transform cdf of
to get
If we now integrate
we have shown that
.
Let F be a cumulative distribution function of a continuous random variable, and
.
Again we use
as in any other transformation:
Now since
is a cdf (e.g.
), take it as a argument of
without any distinctions of its range, which means that in turn
.