This is a very beautiful sample problem from ISI MStat PSB 2013 Problem 8 based on finding the distribution of a random variable . Let's give it a try !!
Cumulative Distribution Function
Normal Distribution
(a) Let be distribution function of X_{2}\) then we can say that ,
=
=
=
Since hence it's symmetric about 0 . So,
and
are identically distributed .
Therefore ,
=
Hence , is also a standard normal random variable.
(b) Let ,
Distribution function
=
= \)
=
=
=
= .
Find the the distribution function of in terms of the cumulative distribution function of a standard normal random variable.
This is a very beautiful sample problem from ISI MStat PSB 2013 Problem 8 based on finding the distribution of a random variable . Let's give it a try !!
Cumulative Distribution Function
Normal Distribution
(a) Let be distribution function of X_{2}\) then we can say that ,
=
=
=
Since hence it's symmetric about 0 . So,
and
are identically distributed .
Therefore ,
=
Hence , is also a standard normal random variable.
(b) Let ,
Distribution function
=
= \)
=
=
=
= .
Find the the distribution function of in terms of the cumulative distribution function of a standard normal random variable.