Theory

Theory 1

Independent random variables

Random variables are independent when they satisfy the product rule for all valid subsets :

Since , this definition is equivalent to independence of all events constructible using the variables and .

For discrete random variables, it is enough to check independence for simple events of type and for and any possible values of and .

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The independence criterion for random variables can be cast entirely in terms of their distributions and written using the PMFs or PDFs.

Independence using PMF and PDF

Discrete case:

Continuous case:

Independence via joint CDF

Random variables and are independent when their CDFs obey the product rule: