To find the marginal PDF fY(y), we need to integrate the joint PDF over x for a given value of y:
fY(y) = ∫ f(x,y) dx
However, the limits of integration for x depend on the value of y:
- If 0 <= y <= 1, then the limits of integration for x are 0 and y.
- If 1 <= y <= 2, then the limits of integration for x are y and 2.
- Otherwise, f(x,y) = 0 for all values of x.
Therefore, we have:
fY(y) = ∫ f(x,y) dx
= ∫{0 to y} a*x^2 dy + ∫{y to 2} a*x^2 dy (if 1 <= y <= 2)
= (1/3)*a*y^3 + (1/3)*a*(2^3 - y^3)
= a*(8/3 - y^3/3) (if 1 <= y <= 2)
= 0 (otherwise)
Thus, the marginal PDF fY(y) is:
- fY(y) = a*(8/3 - y^3/3) if 1 <= y <= 2
- fY(y) = 0 otherwise.
It has the value of a*x^2 if 1 <= x <= 2 and 0 <= y <= x, and 0 otherwise. What's the marginal PDF fY(y)?
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