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Let $Z=X+Y+W$;

where $X∼N(0,σ_1^2)$ i.e. a Gaussian random variable and Y and W follow the Rayleigh distribution:

$f_w(w)=\frac{w}{σ_2^2} . exp(−\frac{w^2}{2σ_2^2})$, $y\ge0$

What will be the distribution of Z, assuming X,Y and W indipendent?

I read another post very similar in this forum: What is the distribution of sum of a Gaussian and a Rayleigh distributed independent r.v.?

and I would only have the confirmation that I could achieve the $Z$'s pdf applying two times the convolution rule. I mean first convolution between $f_y$ and $f_w$, then $f_{yw}$ against $f_x$.

Thanks in advance.

Community
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smtux
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1 Answers1

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Explicit form of the PDF/CDF is difficult to derive. However, the characteristic function of Z can be easily derived from the characteristic functions of X and Y resp. W.

Here we have $$\mathrm{cf}_X(t) = \exp(-\frac{1}{2}\sigma^2_1 t^2),$$ and $$\mathrm{cf}_Y(t) = \mathrm{cf}_W(t) = \ _1F_1\left(1;\frac{1}{2};- \frac{\sigma^2_2}{2}t^2\right) +\sqrt{2}\sigma_2 i t \,_1F_1\left(\frac{3}{2};\frac{3}{2};- \frac{\sigma^2_2}{2}t^2\right).$$ Hence, $$\mathrm{cf}_Z(t) = \exp(-\frac{1}{2}\sigma^2_1 t^2)\left(_1F_1\left(1;\frac{1}{2};- \frac{\sigma^2_2}{2}t^2\right) + \sqrt{2}\sigma_2 i t\, _1F_1\left(\frac{3}{2};\frac{3}{2};- \frac{\sigma^2_2}{2}t^2\right) \right)^2.$$ Analytical inversion of this characteristic function can be difficult. However, numerical inversion is possible, and can be easily performed, e.g. by using the CharFunTool (the Characteristic Functions Toolbox in MATLAB).

Viktor
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