Question: a) Verify that fX,Y,Z is a valid PDF. b) Compute the probability of the event {Z < 1/2} {X > 0} {Y >

Let X, Y, Z be continuous random variables with joint probability density

a) Verify that fX,Y,Z is a valid PDF.

b) Compute the probability of the event {Z < 1/2} ∩ {X > 0} ∩ {Y > 0}.

c) Compute the probability of the event {Z < 1/2} conditional on {X > 0} ∩ {Y > 0}.

d) Compute E[X], E[Y ], and the covariance matrix of X and Y .

Let X, Y, Z be continuous random variables with joint probability density function (PDF) C. (4 + 2x +2y+z), 1 x 1, 1 y 1, 0 z 1, otherwise, fx,y,z(x, y, z)= = where c = = 1/18.

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