Stats. joint densities?
how do i prove this statement?
Suppose that X and Y are random variables with joint density
f (x,y) = 24xy if x>0, y>0, and x+y<1, otherwise 0
X,Y
show that E(Y|X=x) = 2/3(1-x)
What does this tell you about E(Y|X)?
Answers:
I wont give you the entire proof, just give you the key arguments.
knowing the joint distibution of (X, Y) allows to compute the marginal densities of X and Y.
Indeed, let f_X be the marginal density of X, then:
f_X(x) = \int f(x, y) d_y
f_Y(y) = \int f(x, y) d_x
Now, you might compute E(Y| X = x) = \int y f_{Y|x} (y) d_y
where f_{Y| x} is the conditionnal density of Y knowing the event X = x
recall that f_{Y| x} = f(x, y) / f_X(x), and you just have to compute the resulting integrals
regards,
jfpWhy are quadrilaterals such a special group to study? Why not pentagons?
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Suppose that X and Y are random variables with joint density
f (x,y) = 24xy if x>0, y>0, and x+y<1, otherwise 0
X,Y
show that E(Y|X=x) = 2/3(1-x)
What does this tell you about E(Y|X)?
Answers:
I wont give you the entire proof, just give you the key arguments.
knowing the joint distibution of (X, Y) allows to compute the marginal densities of X and Y.
Indeed, let f_X be the marginal density of X, then:
f_X(x) = \int f(x, y) d_y
f_Y(y) = \int f(x, y) d_x
Now, you might compute E(Y| X = x) = \int y f_{Y|x} (y) d_y
where f_{Y| x} is the conditionnal density of Y knowing the event X = x
recall that f_{Y| x} = f(x, y) / f_X(x), and you just have to compute the resulting integrals
regards,
jfp