Are random variables based on the same pmf or pdf always independent?

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Random variables based on the same probability mass function (pmf) or probability density function (pdf) are not always independent. Independence requires that the joint distribution equals the product of the individual distributions, specifically P(A = a, B = b) = P(A=a) * P(B=b). In cases where one variable is a function of another, such as X = Y, the variables are perfectly correlated and thus dependent. To verify independence, one must check the condition of separability in their distributions. Understanding this concept is crucial for accurate statistical analysis.
EdmureTully
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Are they always independent from each other so that you can multiply their E[X] together to form another E[X] with the same distribution and pmf or pdf?
 
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No, in general they need not be independent. In the most extreme case, you may have something like X = some random variable, and Y = X. Then X and Y are 100% correlated, so they are certainly not independent.
 
How do you verify their independence then? Any quick and easy way? Any obvious cases?
 
Hey EdmureTully.

The condition for independence between random variables is P(A = a, B = b) = P(A=a)*P(B=b) where the left hand side is the joint distribution.

If you have this separability, then the two random variables are by definition independent.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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