Probability of 2 Statements (no independence)

But in my case, a is a fraction of b and X is a fraction of Y (again, calculated by a uniform distributed random coefficient between 0.1 and 1.5).

I'm not sure if I'm being daft but isn't it the case that X cannot be normally distributed if X=UY with U and Y as defined above? Since the density function will be given by:

$f_X(x) = \displaystyle\int_{-\infty}^{\infty} f_U(u)f_Y(x/u)\dfrac{1}{|u|} du$ $= (\text{constants}) \displaystyle\int_{0.1}^{1.5} \dfrac{1}{u}\exp \left[-\frac{1}{2}\left(\dfrac{x/u-\mu}{\sigma}\right)^2\right] du$

Which, even in the case of standard normal, doesn't seem to come out much like the PDF of a normally distributed random variable.

[disclaimer: probability is not my thing so I may have missed the point]

Yes, this is true. I expect that what is meant (is this true, OP?) is that X is normally distributed conditional upon the value of u.

Hey guys,
great brainstorming!

In fact, I wanted to keep it simple for everyone. But obviously you are right, my simplicity resulted in a more complex problem.

So X and Y are both normally distributed.
X = u*Y is actually NOT true
E[X] = u*E[Y] and based on those expected values X and Y will be normally distributed.

The question comes from an optimization I want to do, the probability will be in the objective function of a nonlinear problem.