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Answer on Statistics and Probability Question for abid khan

Question #15611
Conditional Probabilities:Select all of the following statements that are true for all joint distributions over X and Y

P(x,y)=P(x)P(y)
P(x,y)=P(x|y)P(y)
P(x,y)=P(x|y)P(y|x)
P(x)=∑yP(x|y)
P(x)=∑yP(x,y)
None of the above.
Expert's answer
Only statement (2) and (5) are true in general.

(1) By definition holds
if X and Y are independent

(2) P(x,y)=P(x|y)P(y) this is definition of
P(x|y):
P(x|y) = P(x,y)/P(y)

(3) the expression P(x|y)P(y|x) has
no sense

(5) ∑yP(x,y) means the sum over all y of values P(x,y). This sum
is equal to P(x)

(4) ∑yP(x|y) means the sum over all y of
P(x|y)=P(x,y)/P(y):

∑yP(x|y) = ∑y P(x,y)/P(y)

again this sum has
no sense

(6) None of the above.

This is also not true, since (2)
and (5) are correct

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