**Mumbai University > Electronics and Telecommunication Engineering > Sem 5 > Random Signal Analysis

Marks: 10M

Year: May 2016

Let $f_{XY}$ (x,y) be joint probability density function of (X, Y)

We introduce auxillary random variable W=Y

Now we have Z=X+Y and W=Y i.e z=x+y and w=y

∴x=z-y =z-w and y=w

∴J=

∴J= =1

We know the joint probability density function $f_{ZW}$ (z,w)=|J| $f_{XY}$ (x,y)=$f_{XY}$ (x,y)

Since X and Y are independent random variables

$f_{XY}$ (x,y)=$f_X$ (x).$f_Y$ (y)

$f_{ZY}$ (z,w)=$f_{XY}$ (x,y)=$f_X$ (x).$f_Y$ (y)

∴$f_{ZW}$ (z,w)=$f_X$ (z-y).$f_Y$ (y)

The marginal probability density function of Z is obtained by integrating $f_{ZW}$ (z,w) w.r.t to w i.e. y

∴$f_Z$ (z)=$∫_{-∞}^∞ {f_X} (z-y).{f_Y} (y)dy$

∴$f_Z$ (z)=$∫_{-∞}^∞{f_Y} (z-x).{f_X} (x)dx$, if W=X