Let (X,Y) be jointly continuous with the following density function.pdf

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Let (X,Y) be jointly continuous with the following density function: f(x,y) = (1/6)e^(-x) where x > 0 and 0 < y < x3. (a) Find the marginal distribution of X. Can you identify it? (b) Find the conditional distribution of Y given X = x. Can you identify it? Solution (a) f(x) =1/6 int (y=0..x^3) e^(-x) dy = 1/6 x^3 e^(-x) Which is a gamma distrubition with parameter k=4 and thetha=1 (gamma(4)=3!=6) (b) f(Y|X=x) = f(x,y)/f(x) = 1/x^3. It seems invalid since integral( 1/x^3 ) for x>0 diverges.

Let (X,Y) be jointly continuous with the following density function:
f(x,y) = (1/6)e^(-x) where x > 0 and 0 < y < x3.
(a) Find the marginal distribution of X. Can you identify it?
(b) Find the conditional distribution of Y given X = x. Can you identify it?
Solution
(a) f(x) =1/6 int (y=0..x^3) e^(-x) dy = 1/6 x^3 e^(-x)
Which is a gamma distrubition with parameter k=4 and thetha=1 (gamma(4)=3!=6)
(b) f(Y|X=x) = f(x,y)/f(x) = 1/x^3.
It seems invalid since integral( 1/x^3 ) for x>0 diverges

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