To determine if X and Y are independent, we need to check if the joint PDF can be factorized into the product of the marginal PDFs of X and Y.
Since the joint PDF at x=1 is 1/2, we have:
fX(x=1) = fX(x=1, y) = ∫ fX,Y(x=1, y) dy
Let's compute the values for fX(x) using the given information:
If 0 < x < 1:
fX(x) = ∫ fX,Y(x, y) dy, where 0 < y < 1
If 1 < x < 2:
fX(x) = ∫ fX,Y(x, y) dy, where 0 < y < 1
To find fY|X(y|x), we can use the conditional probability formula:
fY|X(y|x) = fX,Y(x, y) / fX(x)
If 0 < y < 1/2:
fY|X(y|x=0.5) = fX,Y(x=0.5, y) / fX(x=0.5)
To find fX|Y(x|y), we can use Bayes' theorem:
fX|Y(x|y) = fX,Y(x, y) / fY(y)
If 1/2 < x < 1:
fX|Y(x|y=0.5) = fX,Y(x, y=0.5) / fY(y=0.5)
If 1 < x < 3/2:
fX|Y(x|y=0.5) = fX,Y(x, y=0.5) / fY(y=0.5)
To evaluate E[R|A], we need to calculate the conditional expectation of R given the event A, which is X < 0.5:
E[R|A] = ∫∫ r * fR|A(r|x, y) * fX,Y(x, y) dx dy, where 0 < x < 0.5 and 0 < y < 1
Please note that without the complete joint PDF and additional information, we cannot provide specific numerical values for the above expressions.