PDF of the sum of random variables

[goodmorningda]

Hello everyone, here is the problem I'm currently working on:



I'm stuck on (b). I tried to find the CDF of Y first, is something like this in the ballpark? Thanks!

[goodmorningda]

I just tried again using convolution, I think the case for 0 < y < 1 is correct, but for 1 < y < 2 is a bit messy