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Math Help - Joint cumulative distribution

  1. #1
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    Joint cumulative distribution

    How does one write the joint cumulative distribution

    <br /> <br />
F(X_1 \leq x_1, X_2 \leq x_2) <br /> <br />

    as a combination of marginal distributions?
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  2. #2
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    Hello,

    Quote Originally Posted by peterpan View Post
    How does one write the joint cumulative distribution

    <br /> <br />
F(X_1 \leq x_1, X_2 \leq x_2) <br /> <br />

    as a combination of marginal distributions?
    Have a look here : Marginal distribution - Wikipedia, the free encyclopedia ?


    I think this would lead to :

    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 X_1 and X_2 are discrete random variables ; for continuous ones, see the link, the part with the integrals)
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  3. #3
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    Thanks Moo.

    In this case I am actually considering continuous random variables which are independent, identically distributed.
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  4. #4
    Moo
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    Quote Originally Posted by peterpan View Post
    Thanks Moo.

    In this case I am actually considering continuous random variables which are independent, identically distributed.
    If they are independent, P(X_1 \le x_1, X_2 \le x_2)=P(X_1 \le x_1) \cdot P(X_2 \le x_2)

    Isn't it ?
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  5. #5
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    Thats what I initially thought too, but wasnt sure. Thanks for the clarification!
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  6. #6
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    Is it also fair to say that

    <br /> <br /> <br />
1 - P(X \leq x) = P(X > x)<br /> <br />

    this would then mean that

    <br /> <br />
1 - P(X_1 > x_1, X_2 > x_2) = P(X_1 \leq x_1, X_2 \leq x_2) <br /> <br />
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  7. #7
    Moo
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    Hi again

    Quote Originally Posted by peterpan View Post
    Is it also fair to say that

    <br /> <br /> <br />
1 - P(X \leq x) = P(X > x)<br /> <br />
    Yes

    this would then mean that

    <br /> <br />
1 - P(X_1 > x_1, X_2 > x_2) = P(X_1 \leq x_1, X_2 \leq x_2) <br /> <br />
    No

    Because P(X_1 > x_1, X_2 > x_2) is the simultaneous probability that X_1 > x_1 and X_2 > x_2.

    So its contrary is that X_1 \le x_1 or X_2 \le x_2.
    If you have done a bit of logic, you can understand...

    1 - P(X_1 > x_1, X_2 > x_2) =P(X_1 \le x_1, X_2 > x_2) +P(X_1 \le x_1, X_2 \le x_2)+P(X_1 > x_1, X_2 \le x_2)

    Is it clear ?
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  8. #8
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    Hey Moo

    Im starting to get it...

    Really what im trying to do here is define a joint distribution F(x,y).

    I am trying to do this using the result:

    <br /> <br />
F(x, y) = 1 - P(X > x) - P(Y>y) + P(X > x, Y > y)<br /> <br />

    and the fact that I know the marginal distributions:

    <br /> <br />
P(X \leq x), P(Y \leq y) <br /> <br />

    Still a little unclear on how this works.

    Thanks

    Peter
    Last edited by peterpan; June 7th 2008 at 11:21 AM.
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