Let $X$ have a standard normal distribution and $Y$ have an Exponential(1) distribution, with $X$ and $Y$ being independent random variables. Find the density function of Φ(X)+Y , where Φ denotes the standard normal distribution function.
I am attempting to do this problem through convolution. However, I am not sure about the density of the Φ(X) in the first place, but I know that the density of $X$ is $\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}$, is there a way to use this fact to find density of Φ(X)? Thank you.
The density of $Z=\Phi(X)$ is uniform in$(0;1)$ and it is easy to prove
Proof:
Let $Z=F_X(x)$ with F continuous.
$$F_Z(z)=P[Z \leq z]=P[F_X(x)\leq z]=P[X\leq F_X^{-1}(z)]=F_X[F_X^{-1}(z)]=z$$
Thus
$$f_Z(z)=\mathbb{1}_{(0;1)}(z)$$