A random variable has Gaussian distribution with mean zero and variance one, while a random variable has the distribution with the density:
What is the distribution of assuming that and are independent?
One approach is to use the formula of change of variables (e.g. using Jacobian) for a product of random variables (see for example Grimmett and Stirzaker, p. 109) through the map: which gives:
, use independence and integrate over to obtain the result.
But the formula is somewhat cumbersome as it gives 2 different expressions whether you assign or to (no sign problem with ). I assume you get to the same result although the calculation doesn't seem tractable.
Another (simpler?) approach is to use the formula of total probability and use independence. But again it doesn't seem you can go beyond an integral of the form:
Is this correct? Is it possible to do better?
Thanks for any help.