Hey guys,
Consider the following short example of transformations.
Let the joint density of \(\displaystyle X\ \text{and}\ Y\) be given by the unit square, i.e.
\(\displaystyle \displaystyle f_{X,Y}\left (x,y \right) = \begin{cases} 1\ 0<x<1\ \text{and}\ 0<y<1 \\ \text{elsewhere} \end{cases}\)
Then the Cumulative Distribution Function of \(\displaystyle Z=X+Y\) is given by:
\(\displaystyle \displaystyle F_Z \left( z \right)= \begin{cases}0\ \text{for}\ z<0 \\ \int_0^{z} \int_0^{z-x} dydx\ \text{for}\ 0\leq z <1 \\1-\int_{z-1}^1 \int_{z-x}^1 dydx\ \text{for}\ 1\leq z<2 \\1\ \text{for}\ 2\leq{z} \end{cases}\)
I understand why we have to partition our CDF, what I am having trouble figuring out is why for the interval \(\displaystyle \left[ 1,2 \right) \) that specific form. What is the intuition here? Thanks.
Consider the following short example of transformations.
Let the joint density of \(\displaystyle X\ \text{and}\ Y\) be given by the unit square, i.e.
\(\displaystyle \displaystyle f_{X,Y}\left (x,y \right) = \begin{cases} 1\ 0<x<1\ \text{and}\ 0<y<1 \\ \text{elsewhere} \end{cases}\)
Then the Cumulative Distribution Function of \(\displaystyle Z=X+Y\) is given by:
\(\displaystyle \displaystyle F_Z \left( z \right)= \begin{cases}0\ \text{for}\ z<0 \\ \int_0^{z} \int_0^{z-x} dydx\ \text{for}\ 0\leq z <1 \\1-\int_{z-1}^1 \int_{z-x}^1 dydx\ \text{for}\ 1\leq z<2 \\1\ \text{for}\ 2\leq{z} \end{cases}\)
I understand why we have to partition our CDF, what I am having trouble figuring out is why for the interval \(\displaystyle \left[ 1,2 \right) \) that specific form. What is the intuition here? Thanks.