A Distribution functions

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mathman

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Summary
Marginal distributions for multivariable
Let F(x,y) be the joint distribution for random variables X and Y (not necessarily independent). Is ##lim_{y\to \infty}F(x,y)## the distribution function for X? I believe it is. How to prove it?
 

StoneTemplePython

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Summary: Marginal distributions for multivariable

Let F(x,y) be the joint distribution for random variables X and Y (not necessarily independent). Is ##lim_{y\to \infty}F(x,y)## the distribution function for X? I believe it is. How to prove it?
This is rather indirect, but I felt like answering this with inequalities, using a thread you were on earlier this year.
ref: https://www.physicsforums.com/threads/joint-pdf-and-its-marginals.965416/

Upper Bound
##F_{X,Y}(x,y)\leq F_X(x) ##

- - - -
with ##A## the event ##X \leq x## and ##B## the event ##Y \leq y##

##F_{X,Y}(x,y)##
##= P\big(A \cap B\big) ##
##\leq P\big(A \cap B\big) + P\big(A \cap B^C\big) ##
##= P\big( (A \cap B) \cup (A \cap B^C)\big ) ##
##= P\big( A\big ) ##
## =F_X(x)##

note: the upper bound does not depend on choice of y


Lower Bound (see link)
##F_X(x) + F_Y(y) - 1 \leq F_{X,Y}(x,y) ##

putting upper and lower bounds together and taking limits, recalling that Y is a bona fide random variable, so you know its CDF tends to one

## F_X(x) ##
##= \big\{ F_X(x) + 1 - 1\big\} ##
##= \lim_{y \to \infty}\big\{ F_X(x) + F_Y(y) - 1\} ##
##\leq \lim_{y \to \infty} F_{X,Y}(x,y) ##
##\leq F_X(x)##

as desired
 
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