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

Marks: 10M

Year: May 2016

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=zy=zw and y=w ∴J=

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

Since X and Y are independent random variables

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

Since X and Y are independent random variables

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

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

∴$f_{ZW}$ (z,w)=|y|$f_X$ (yz).$f_Y$ (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)=$∫_{-∞}^∞ |y|f{_X} (yz).{f_Y} (y)dy$

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