Joint probability distribution function of W=Y-X Let $X$ and $Y$ denote the arrival times of the first two calls at a telephone switch.the joint pdf of $X$ and $Y$ is:
$$f_{X},_{Y}(x,y)=\begin{cases}
\lambda ^{2}e^{-\lambda y} & \text{  }0\leqslant  x< y \\ 
0 & \text{ otherwise } 
\end{cases}$$
A) what is the PDF of W=Y-X ? 
B) Are $W$ and $X$ independent? 
C) Are $W$ and $Y$ independent?
part A) for w<0  or  x>y $F_{W}(w)=0 $
for w>0 
http://i.imgsafe.org/b7bdf29.jpg
$$
\int_{w}^{\infty}\int_{0}^{y}x\lambda ^{2}dxdy=\lambda ^{2}\int_{w}^{\infty}xe^{-\lambda y}]_{0}^{y}dy=\lambda ^{2}\int_{w}^{\infty}ye^{-\lambda y}=\lambda ^{2}(\frac{-ye^{-\lambda y}}{\lambda }-\frac{e^{-\lambda y}}{\lambda ^{2}}) ]_{w}^{\infty}=\lambda ^{2}(0-(\frac{-we^{-\lambda w}}{\lambda }-\frac{e^{-\lambda w}}{\lambda ^{2}})) 
$$
Is my solution for part A correct?
part B&C)the joint pdf of W and X or Y is not given how to show they are independent or not?any ideas would be appreciated.
$$ f_{W},_{X}(W,X)=f_{W}(W)*f_{X}(X) $$
$$ f_{W},_{X}(W,X)=? $$
 A: a) It's easy to find the PDF for $X$ and $Y$, we see
$$ f_X (x) = \int_0^\infty f_{X,Y}(x,y) dy =  \lambda^2 \int_x^\infty e^{ - \lambda y}  dy = \lambda e^{ - \lambda x} $$
$$ f_Y(y) = \int_0^\infty  f_{X,Y} (x,y) dx = \lambda^2e^{-\lambda y}  \int_0^y dx = \lambda^2 y e ^{ - \lambda y} $$
Note that we see that CDF of $W$ is, (using the joint PDF)
$$F_{W}(w)= \mathbb{P} (  w \geq W=Y-X ) = \mathbb{P} ( w+X \geq Y) =  \int_0^\infty \int_x^{w+x} f_{X,Y}(x,y)dydx = $$
$$= \lambda \int_0^\infty (e^{-\lambda x} - e ^{ - \lambda( x + w) } ) dx = 1 - e^{ - \lambda w }  $$
Thus the PDF of $W$ is just
$$ f_W (w ) =  \lambda e^{-\lambda w}  $$
b) I won't do the computation for b,c, but to find out if $W$ and $X$ are independent. We need to check
$$ \mathbb{P} (\tilde w \geq W , \tilde x \geq X ) = \mathbb{P} ( \tilde w \geq W ) \mathbb{P} ( \tilde x \geq X) $$ 
Looking at the above computation, we see this amounts to checking
$$ \int_0^{\tilde x} \int_x^{w+x} f_{X,Y}(x,y) dydx = \int_0^{ \tilde w} f_W(w) dw \int_0^{ \tilde x} f_X(x) dx$$
Repeat for c).
