f (x, y) is always equal to or greater than 0 for every (x, y) in the domain of X and Y.

The integration of f (x, y) over the entirety of the domain of X and Y equates to 1.

In mathematical terms, the second condition can be articulated as:

∬ f (x, y) dx dy = 1

f_X (x) = ∫ f (x, y) dy

f_Y (y) = ∫ f (x, y) dx

f_X|Y (x|y) = f (x, y) / f_Y (y)

f_Y|X (y|x) = f (x, y) / f_X (x)

f (x, y) = f_X (x) * f_Y (y)

If this condition is satisfied, the variables do not impact each other, and knowledge of one variable does not yield any information regarding the other variable.

Risk assessment: Modeling the joint probabilities of several risk factors to gauge their combined effect on a system or an investment.

Engineering: Analyzing the dependencies between multiple components or variables in a system to enhance its performance.

Economics: Examining the relationships between economic variables, including inflation and unemployment, to comprehend their joint behavior and formulate economic policies.

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