Discrete and Continuous Random Variables: How to find the PDF/CDF?

I have a R.V. Z = { toss of a fair coin to decide X or Y}


X represents a number of years and is a continuous Uniform R.V. in interval [a,b]


Y represents a number of years and is a continuous Uniform R.V. in interval [c,d]





If I need to find the PDF and CDF of the number of years that I could get in this scenario.. do I multiply the PDF of X by 1/2 (prob of getting X with a fair coin) and the same for Y? or am i combining the PDFs of X and Y to get a joint PDF..? I'm not sure what to do with these three different PDFs/CDFs

Discrete and Continuous Random Variables: How to find the PDF/CDF?
You are correct:


pdf of Z = 1/2 x pdf of X + 1/2 x pdf of Y


so


f(z) = 1/[2(b-a)] + 1/[2(d-c) ] if z is inside [a,b] ànd inside [c,d]


f(z) = 1/[2(b-a)] if z is inside [a,b] and outside [c,d]


f(z) = 1/[2(d-c)] if z is inside [c,d] and outside [a,b]


f(z) = 0 if z is outside [c,d] and outside [a,b]





To find cdf there are several case to be considered, depending on the position of a,b,c and d





Suppose for example that the intervals are not overlapping and that b %26lt; c


In this case we have


f(z) = 0 if z %26lt; a]


f(z) = 1/[2(b-a)] if a %26lt;z %26lt; b


f(z) = 0 if b %26lt; z %26lt; c


f(z) = 1/[2(d-c)] if c %26lt; z %26lt; d


f(z) = 0 if z %26gt; d


and then cdf


F(z) = 0 if z %26lt; a


F(z) = (z-a)/[2(b-a))] if a %26lt; z %26lt; b


F(z) = 1/2 if b %26lt; z %26lt;c


F(z) = 1/2 + (z-c)/[2(d-c)] if c %26lt; z %26lt; d


F(z) = 1 if z %26gt; d





Other cases are


- [c,d] is included in [a,b]


- [c,d] and [a,b] overlap