You still need to integrate over x. y|x=c is Gaussian for each c, but there are many such c's in [a,inf]. If you're curious: https://www.ams.org/journals/mco

You still need to integrate over x. y|x=c is Gaussian for each c, but there are many such c's in [a,inf]. If you're curious: https://www.ams.org/journals/mcom/1978-32-141/S0025-5718-1978-0461849-9/S0025-5718-1978-0461849-9.pdf https://www.ams.org/journals/mcom/1978-32-141/S0025-5718-197...