A question about the conditional expected value

Question

The question is here,

Let R = XY and let A be the event X< 0.5. Evaluate E[R |A].

And I know the the fX(x)

The given answer is here: The event A leaves us with a right triangle with a constant height. The conditional PDF is then 1/area =8. Theconditional expectationyields:

And my solution is below:

There seems some difference between my answer and given answer... Why am I wrong? I check it for several times and do not know...

Thanks your help! I don't know how the type math symbols.. Sorry..

Answer 1

I believe the problem is that you are using the formula for a conditional probability where the condition is $X=x$. However, the condition here is that $X<.5$. Therefore, instead of using $x/2$ which is the $f(x)$ for a given $x$, use 1/16 which is the probability that $x<.5$. This should get you the correct answer.

Answer 2

You may miss $f_{X|A}(x|A)$, which is $8x$. Note that $E[XY|A]=\int_{0}^{0.5}{E[XY|X=x]f_{X|A}(x|A)dx}$.