Let X_1, X_2, X_3 are independent random variables and let the probability density of each variable be as following
$$f(x_i) = e^{-x}I_{(0,\infty)}(x)$$
Now, First I would like to derive joint probability density function of $Y_1 = X_1, Y_2 = X_1+X_2, Y_3 = X_1+X_2+X_3$
my solution
the domain of each $Y_i$ is $(0,\infty)$ by the definition of each random variable and the domain of each $X_i$
Then since joint probability of $X_1, X_2, X_3$ is $g(x_1,x_2,x_3) = e^{-x_1-x_2-x_3}I_{(0<x_i<\infty, \;i = 1,2,3)}$ by independence between $X_i$
Then if $X_i$ are substituted with random variables $Y_i$ by $X_1 = Y_1, X_2 = Y_2-Y_1, X_3 = Y_3 - Y_1- Y_2$, $h(y_1,y_2,y_3) = e^{y_1-y_3}I_{(0<y_1, y_3\infty)}$
Second, I need to derive joint pdf of $Y_1$ and $Y_3$
Since $X_3 = Y_3-Y_1-Y_2$ One can derive $f(y_3-y_1-y_2) = e^{-y_3+y_1-y_2}I_{(0<y_i, \;i=1,2,3<\infty)}$ then the marginal probability about y_2 would be $\int_0^\infty e^{-y_3+y_1-y_2}I_{(0<y_i, \;i=1,2,3<\infty)}dy_2=e^{y_1-y_3}I_{(0<y_1, y_3<\infty)}$
From the first and second, it is revealed that joint pdf $Y_1, Y_2, Y_3$ equals to its joint pdf of $Y_1$ and $Y_3$
(I don't know what it means more mathematically anyway, in other words any specific term or expression we refer to upon this characteristic?)