Math Help - Pdf of minimum of (x,c)

I have to find the pdf of y = minimum of (x,c) where is a Random Variable with exponential distribution and ' ' is a constant. The detailed steps of my approach was given in the attachment. Is this approach correct? Please comment on the correctness of the solution

To be sure, I tried solving it myself, and I get a different result.

Check it on:

https://docs.google.com/document/pub...tpnmGR2yr9Osn0

Strictly speaking, Y doesn't admit a pdf. There is an atom at c: or according as or (note both are constants) so there is an error in this step in your work.

You can do this with the dirac delta function if you're comfortable with that. It looks like rargh has the right answer.