What is a Marginal Distribution and How Does it Apply to F1(x)F2(y)?

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SUMMARY

A marginal distribution represents the probability distribution of a subset of variables within a larger multivariate distribution. In the context of independent random variables X and Y, the joint cumulative distribution function I(x,y) can be expressed as I(x,y) = F1(x)F2(y), where F1 and F2 are the marginal distributions of X and Y, respectively. This relationship holds true under the condition that X and Y are independent, allowing for the simplification of joint probabilities into products of individual probabilities.

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I'm not sure I understnad what is a marginal distribution, but i need to show that if F1,F2 are one dimensional cummulative distribution functions then I(x,y)=F1(x)F2(y) has F1 and F2 as its marginal distributions.

well if I(x,y)=P(X<=x,Y<=y) and if X and Y are independent, then it equals: P(X<=x)*P(Y<=y), then F1(x)=P(X<=x) F2(y)=P(Y<=y)
or in general: F1(x)=P(X<=x, Y[tex]\in[/tex]A) F2(y)=P(X [tex]\in[/tex] B Y<=y) where A and B are intervals where the r.vs Y and X are defined.

but it's really a guess.
 
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What is the definition of a marginal distribution? How does that def. apply to F1(x)F2(y)?
 

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