I'm trolling through some old papers and this particular question has me stumped on how to solve - specifically I'm unsure how to proceed to derive the pdf of Y given that X is continuous while U is discrete.

Q: "Let X be an exponential random variable (pdf: ) and U a random variable that takes values +/- 1 with probabilities 0.5 each. Find the pdf of Y=XU."

Advice gratefully received!