How does one write the joint cumulative distribution
$\displaystyle
F(X_1 \leq x_1, X_2 \leq x_2)
$
as a combination of marginal distributions?
Hello,
Have a look here : Marginal distribution - Wikipedia, the free encyclopedia ?
I think this would lead to :
$\displaystyle P(X_1 \le x_1, X_2 \le x_2)=\sum_{k \le x_2} P(X_1 \le x_1, X_2=k)=\sum_{k \le x_2} \ \sum_{l \le x_1} P(X_1=l, X_2=k)$
(assuming that $\displaystyle X_1$ and $\displaystyle X_2$ are discrete random variables ; for continuous ones, see the link, the part with the integrals)
Hi again
Yes
Nothis would then mean that
$\displaystyle
1 - P(X_1 > x_1, X_2 > x_2) = P(X_1 \leq x_1, X_2 \leq x_2)
$
Because $\displaystyle P(X_1 > x_1, X_2 > x_2)$ is the simultaneous probability that $\displaystyle X_1 > x_1$ and $\displaystyle X_2 > x_2$.
So its contrary is that $\displaystyle X_1 \le x_1$ or $\displaystyle X_2 \le x_2$.
If you have done a bit of logic, you can understand...
$\displaystyle 1 - P(X_1 > x_1, X_2 > x_2) =P(X_1 \le x_1, X_2 > x_2)$$\displaystyle +P(X_1 \le x_1, X_2 \le x_2)+P(X_1 > x_1, X_2 \le x_2)$
Is it clear ?
Hey Moo
Im starting to get it...
Really what im trying to do here is define a joint distribution $\displaystyle F(x,y)$.
I am trying to do this using the result:
$\displaystyle
F(x, y) = 1 - P(X > x) - P(Y>y) + P(X > x, Y > y)
$
and the fact that I know the marginal distributions:
$\displaystyle
P(X \leq x), P(Y \leq y)
$
Still a little unclear on how this works.
Thanks
Peter