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Unformatted text preview: plane then the expectation of the random variable g (X, Y ) can be computed using the law of the unconscious statistician (LOTUS), just as for functions of a single random variable: ∞ ∞ E [g (X, Y )] = g (u, v )fX,Y (u, v )dudv. −∞ −∞ The following two properties are necessary and sufficient for f to be a valid joint pdf: Property pdf.1: f is nonnegative Property pdf.2: R2 f (u, v )dudv = 1. Property pdf.1 has to be true for any pdf f because, by (4.2), the integral of f over any set is the probability of an event, which must be nonnegative. Property pdf.2 has to be true because the integral shown is, by (4.2), equal to P {(X, Y ) ∈ R2 }, which must equal one because, by definition, the pair (X, Y ) always takes a value in R2 . Note that pdfs are defined over the entire plane R2 , although often the pdfs are zero over large regions. The support of a joint pdf is the set over which it is nonzero. The pdf of X alone or of Y alone can be obtained from the joint pdf, as follo...
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