Past exam of the options I course of the University of Oxford / 2012 / M / M2 / a

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- Solution a

Solution

A joint probability density function must be measurable, nonnegative almost everywhere, and normalized:

f (x, y) \geq 0 almost everywhere, \int_{R^{2}} f (x, y) d x d y = 1.

It determines the joint law by

Pr ((X, Y) \in B) = \int_{B} f (x, y) d x d y

for measurable sets

B

. Values on sets of planar measure zero may be changed without changing that law.

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