This is a very beautiful sample problem from ISI MStat PSB 2008 Problem 7 based on finding the distribution of a random variable . Let's give it a try !!
Let and
be exponential random variables with parameters 1 and 2 respectively. Another random variable
is defined as follows.
A coin, with probability p of Heads (and probability 1-p of Tails) is
tossed. Define by
Find
Cumulative Distribution Function
Exponential Distribution
Let , be the CDF for i=X,Y, Z then we have ,
= =
Therefore pdf of Z is given by , where
are pdf of X,Y respectively .
So ,
Find the the distribution function of and then find
This is a very beautiful sample problem from ISI MStat PSB 2008 Problem 7 based on finding the distribution of a random variable . Let's give it a try !!
Let and
be exponential random variables with parameters 1 and 2 respectively. Another random variable
is defined as follows.
A coin, with probability p of Heads (and probability 1-p of Tails) is
tossed. Define by
Find
Cumulative Distribution Function
Exponential Distribution
Let , be the CDF for i=X,Y, Z then we have ,
= =
Therefore pdf of Z is given by , where
are pdf of X,Y respectively .
So ,
Find the the distribution function of and then find