Question 1172515
.
Suppose that  the joint density function of  the random variables X and Y is given by 
         F(x,y) ={8xy if 0≤y≤x≤1
                   0     elsewhere
a)ComputeP(X+Y <1).(Sketch the region clearly.)
b)Find the marginal density of X, i.e.fX(x) and marginal density of Y, i.e.fY(y).
c)Find the conditional density of Y given X=1/2, that is,fY|X(y|1/2).
d)Find the conditional expectation of Y given X=1/2, that is,E[Y|X=1/2].
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In the post by @CPhill, the region for integration is determined INCORRECTLY, 
therefore, all his subsequent calculations are INCORRECT and IRRELEVANT.


Indeed, at the end of n.1, @CPhill writes


<pre>
    * The intersection is a triangle with vertices (0, 0), (1, 0), and (1/2, 1/2).
</pre>


It is INCORRECT.  Actually, this triangle has vertices (0,0), (1,0) and (0,1).


As I said, everything that follows in the post by @CPhill is WRONG.



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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Regarding the post by @CPhill . . . 



Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.


The artificial intelligence is like a baby now. It is in the experimental stage 
of development and can make mistakes and produce nonsense without any embarrassment.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It has no feeling of shame - it is shameless.



This time, again, &nbsp;it made an error.



Although the @CPhill' solution are copy-paste &nbsp;Google &nbsp;AI solutions, &nbsp;there is one essential difference.


Every time, &nbsp;Google &nbsp;AI &nbsp;makes a note at the end of its solutions that &nbsp;Google &nbsp;AI &nbsp;is experimental
and can make errors/mistakes.


All @CPhill' solutions are copy-paste of &nbsp;Google &nbsp;AI &nbsp;solutions, with one difference:
@PChill never makes this notice and never says that his solutions are copy-past that of Google.
So, he NEVER SAYS TRUTH.


Every time, &nbsp;@CPhill embarrassed to tell the truth.

But I am not embarrassing to tell the truth, &nbsp;as it is my duty at this forum.



And the last my comment.


When you obtain such posts from @CPhill, &nbsp;remember, &nbsp;that &nbsp;NOBODY &nbsp;is responsible for their correctness, 
until the specialists and experts will check and confirm their correctness.


Without it, &nbsp;their reliability is &nbsp;ZERO and their creadability is &nbsp;ZERO, &nbsp;too.