# Thread: Joint Density f(X,Y) integration when there is no Y in the eqn

1. ## Re: Joint Density f(X,Y) integration when there is no Y in the eqn

Ah ok, thank you! I'm on to part

(c) Find E(X), E(Y ), E(Y |X) and V ar(X|Y ).

I get the following for E(X) which I think is correct:

But when I do it for E(y) I run into a problem on the 6th step:

I don't know what to do with the X in a(1/2 - x1/2 - 0 - 0). I need to solve for a single number for E(y) but the x is giving me problems.

2. ## Re: Joint Density f(X,Y) integration when there is no Y in the eqn

You still have not get the correct marginal pdf for .

Note

is independent of x, as you have already integrate the dummy variable out.

Actually, from the given joint density, without doing the integration you
can immediately tell
and is independent from .

3. ## Re: Joint Density f(X,Y) integration when there is no Y in the eqn

Hmm. How do I have the X value already out? Im not really following.

4. ## Re: Joint Density f(X,Y) integration when there is no Y in the eqn

Do you know the uniform distribution?

Or more directly, can you get the result

5. ## Re: Joint Density f(X,Y) integration when there is no Y in the eqn

Ok, firstly, can be seen to be uniform before any work is completed?

Which leads to my next question, how can I deduce that

Actually, from the given joint density, without doing the integration you
can immediately tell
and is independent from .
What is the process to come to that result?

Lastly, would the marginal distribution of Y then simply be

6. ## Re: Joint Density f(X,Y) integration when there is no Y in the eqn

Originally Posted by crusoe
Ok, firstly, can be seen to be uniform before any work is completed?
The joint density isn't uniform. The marginal of Y is uniform. The joint depends on the value of x so it can't be uniform.

Lastly, would the marginal distribution of Y then simply be
This can't be a marginal distribution for Y. Why not? Well it's a function of x... and the marginal distribution needs to be a function of y only.

Edit: Also, you already know what 'a' is. Stop referring to 'a' and instead replace any reference of 'a' with 2.

7. ## Re: Joint Density f(X,Y) integration when there is no Y in the eqn

So what would the marginal distribution for Y be then? I think BGM confused me more.

8. ## Re: Joint Density f(X,Y) integration when there is no Y in the eqn

lol sorry to confuse you.

Actually I think that in the process of solving the normalizing constant ,
you should already have a rough idea about the marginal distributions of
and respectively.

Anyway, let see if you agree with the following steps. You are almost got
everything correct (except some places do not substitute back, I do not know why).
Just reply if you cannot accept/agree with any one of them.

1. The solution

2. Relationship between the joint distribution and marginal distribution:

3. The given joint distribution:

4. Combining 2 and 3 gives

5. Integration of polynomials:

6. Integration is linear:

7. Combining 5 and 6 gives

8. So the conclusion is
and

If you are familiar with the definition of independence,
then you should know are independent if and only if

provided that their joint and marginal density exists.

So by observations, if the joint density can be written as
for some functions
independent of and independent of ,
(of course the support is somehow "rectangular", independent of ,
e.g. in this case)
then we know that
and are independent.

Thats the reason I say that by observation, without really doing integration,
you should be confident to say they are independent. Also, since the joint
distribution is independent from the dummy variable of ,
the marginal pdf of is a constant function (on [0, 1]),
so the marginal distribution of must be a uniform distribution.

9. ## Re: Joint Density f(X,Y) integration when there is no Y in the eqn

That all makes sense, thank you. So back to my question regarding E(Y)

(c) Find E(X), E(Y ), E(Y |X) and V ar(X|Y ).

I get the following for E(X) which I think is correct:

But when I do it for E(y) I run into a problem on the 6th step:

I don't know what to do with the X in a(1/2 - x1/2 - 0 - 0). I need to solve for a single number for E(y) but the x is giving me problems.

10. ## Re: Joint Density f(X,Y) integration when there is no Y in the eqn

Ok. So back to the problem.

Do you agreed that

1.

2.

3.