I'm currently self-studying and I'm looking for books focusing on random variables and their transformations, which possibly contain examples like the one in this question.
I'm also interested in convolution of distributions.
Any books you've read and would recommend? Thanks.
Handbook on Statistical Distributions for Experimentalists - University of Stockholm
A link to the pdf http://www.stat.rice.edu/~dobelman/textfiles/DistributionsHandbook.pdf
Cambridge University posts lectures notes and homework online. You could visit there. There is a list of courses available here http://www.cl.cam.ac.uk/teaching/1415/
Transforming Density Functions - https://www.cl.cam.ac.uk/teaching/2003/Probability/prob11.pdf
Convultion of distributions
Here are some lecture notes I found from MATH 526 University of Washington Lecture 8 will focus upon convolution. This link will take you to all the lectures and homework.
http://www.math.washington.edu/~hart/m526/
http://www.math.washington.edu/~hart/m526/Lecture8.pdf
University of Manchester - Products and Convolutions of Gaussian Probability Density Functions
http://www.tina-vision.net/docs/memos/2003-003.pdf
If you could be more specific I am sure I could find more.