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Find the MGF of an r.v. that is a function of a continuous r.v. & a discrete r.v.

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: Attachment 24055) and U a random variable that takes values +/- 1 with probabilities 0.5 each. Find the pdf of Y=XU."

Advice gratefully received!

Re: Find the MGF of an r.v. that is a function of a continuous r.v. & a discrete r.v.

hint: $\displaystyle P(Y \leq k) = P(Y \leq k \cap U=-1) + P(Y \leq k \cap U=1)$

Re: Find the MGF of an r.v. that is a function of a continuous r.v. & a discrete r.v.

You don't say it but I assume that the two random variables are independent.