Let $W_r$ denotes time taken for the r-th occurrence of the phenomenon in Poisson process $\{N_t : t \ge 0\}$ with occurrence rate $\lambda$
$$W_r = \min\{t:N_t \ge r\},\; r= 1,2,3..$$
Here I want to derive joint pdf of $X = W_2/W_4, Y = W_4/W_5$.
First for the random variable $X= W_2/W_4,$ since $W_2 \thicksim Gamma(2, 1/\lambda)$ and $W_4 \thicksim Gamma(4, 1/\lambda)$, $f(w_2)=\dfrac{1}{\Gamma(2)(1/\lambda)^4}x\cdot e^{-\lambda x}I(x>0)$ and $f(w_4)=\dfrac{1}{\Gamma(4)(1/\lambda)^4}x\cdot e^{-\lambda x}I(x>0)$.
Then $f(X=W_2/W_4) = \dfrac{\Gamma(4)}{\Gamma(2)}\cdot\lambda^2=6\lambda^2$
But though I had derived pdf of $X$,when I integrate it through $0$ to $\infty$ it doesn't result in $1$ and I think there I had done some wrong.
Any hint or advice to approach this problem correctly?