Let $Z=X+Y$; where $X\sim \mathscr N(0,\sigma^2_1)$ i.e. a Gaussian random variable and $Y$ follows the Rayleigh distribution: $$ f_Y(y) = \frac{y}{\sigma^2_2}\exp\left(-\frac{y^2}{2\sigma^2_2}\right) \mathbf{1}_{y \geqslant 0} $$ What will be the distribution of $Z$?
What is the distribution of sum of a Gaussian and a Rayleigh distributed independent r.v.?
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probability-distributions
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0Would you like to know if the distribution of $x+y$ belongs to a known class of distributions? Also: are $x,y$ independent? – 2012-05-09