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.