3. Let the joint probability density function for ( X , Y ) be f ( x, y ) = C x y , x > 0, y > 0, 3 x + y < 3,
a) Sketch the support of ( X , Y ).
b) Find the value of C so that f ( x, y ) is a valid joint p.d.f.
c) Find the probability P ( X < Y ).
d) Find the marginal probability density function of X, f X ( x ).
e) Find the marginal probability density function of Y, f Y ( y ).
4. Let the joint probability density function for ( X , Y ) be f ( x, y ) = C x 2 y , zero otherwise. That is, sketch { x > 0, y > 0, 3 x + y < 3 }. 0 < x < 1, x < y < 2, zero otherwise.
a) Sketch the support of ( X , Y ).
b) Find the value of C so that f ( x, y ) is a valid joint p.d.f.
c) Find the marginal probability density function of X, f X ( x ).
d) Find the marginal probability density function of Y, f Y ( y ).
e) Are X and Y independent? Justify your answer. No credit will be given without
proper justification. That is, sketch { 0 < x < 1, x < y < 2 }.
proper justification. That is, sketch { 0 < x < 1, x < y < 2 }.
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