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How to determine the transformation function between two given random variables

Q: "If a random variable has the pdf : Attachment 24053 find the transformation that would generate this random variable from the standard uniform."

My approach is laid out below, but it's a fail since if I then try to use the transform function against an r.v. that's 'standard uniform' (ie: Y~U[0,1]), I'll generate the pdf for X as above just fine, but the interval is borked (ie: I don't get -1<= x <= 1). What am I missing here?

Attachment 24054<click image to enlarge to human-readable size>

Re: How to determine the transformation function between two given random variables

the limits on your left hand integral are incorrect. you need to integrate the pdf between -1 and x.

ie, solve: