Transformation theorem
The transformation theorem concerns the expected value of a function of a
random variable. If
is a random variable and
![[eq1]](http://images2.statlect.com/transformation_theorem__2.png)
is a function, then the expected value of the random variable
![[eq2]](http://images1.statlect.com/transformation_theorem__3.png)
can
be computed without knowing the distribution of
.
For example, when
is a continuous random variable, we can compute the expected value of
as![[eq3]](http://images1.statlect.com/transformation_theorem__7.png)
where
![[eq4]](http://images2.statlect.com/transformation_theorem__8.png)
is the probability density function of
,
instead of computing it
as![[eq5]](http://images2.statlect.com/transformation_theorem__10.png)
where
![[eq6]](http://images1.statlect.com/transformation_theorem__11.png)
is the probability density function of
.
More details about the transformation theorem can be found in the lecture entitled Expected value.
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