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?

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- Jun 7th 2008, 05:17 AMpeterpanJoint cumulative distribution
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? - Jun 7th 2008, 05:24 AMMoo
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) - Jun 7th 2008, 05:35 AMpeterpan
Thanks Moo.

In this case I am actually considering continuous random variables which are independent, identically distributed. - Jun 7th 2008, 05:39 AMMoo
- Jun 7th 2008, 06:06 AMpeterpan
Thats what I initially thought too, but wasnt sure. Thanks for the clarification!

- Jun 7th 2008, 07:47 AMpeterpan
Is it also fair to say that

$\displaystyle

1 - P(X \leq x) = P(X > x)

$

this 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)

$ - Jun 7th 2008, 08:03 AMMoo
Hi again :)

Yes

Quote:

this 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 ? (Wink) - Jun 7th 2008, 09:00 AMpeterpan
Hey Moo (Hi)

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 (Cool)