If X and Yare jointly distributed random variables with cont

If X and Yare jointly distributed random variables with continuous joint density function f_x,y(x, y) show that their sum Z = X+ Y is continuous with density function fz(z) = Integral_-infinity^infinity f_x,y(z, t - z)dx.

Solution

X and Y are continous and the density funcion fx,y

so the sum of these variable will be continous too

Z = X+Y

the pdf will be

Fz (z) = fxy ( z,t-z) dx

where t is a auxiliary variable