I'd like to calculate the PDF of the sum of two random variables one following an Arcsine distribution and the other a Gaussian one, using the convolution. After some manipulations of the integral, I obtain $\int _ { 0 } ^ { \pi } \exp - \frac { ( z - A \cos ( \alpha ) ) ^ { 2 } } { 2 \sigma ^ { 2 } } d \alpha$ and can't go ahead.
result of the convolution of the Arcsine and the Gaussian distributions
A, z and sigma are constant for the integral. The random variables X and Y follow respectively: $f _ { X } ( x ) = \frac { 1 } { \pi \sqrt { A ^ { 2 } - x ^ { 2 } } }$ $f _ { Y } ( y ) = \frac { 1 } { \sigma \sqrt { 2 \pi } } \exp \left( - \frac { y ^ { 2 } } { 2 \sigma ^ { 2 } } \right)$
X and Y distributions Please can someone give me tips to go ahead, thanks in advance
