Use the conditions you have.
Now what are your boundaries for
What can you infer about the bounds here?
Let me know if you have questions.
Let X and Y have the joint probability distribution given by
p(x,y) = exp(-(x+y)) for x>0 and y>0
what is P(X+Y<3)?
i know that i would need to use double integrals for dx and dy but i do now know what i can put as the range of x and y such that the condition is fulfilled.
for what values of Y is the conditional density p(x l y) defined?
Basically what I did was use the facts you were given:
Then, I wrote in terms of and we know since if our condition is automatically not satisfied. So we obtain:
For we integrate over integrate over
The first integral is usually straightforward. can take on values from So you have:
Hey, you erased your original question?????
So you are given already. Now we just need to find the marginal density in order to determine the conditional density. Then we can use the stipulations on and to determine for what values of and the conditional density is defined.
The marginal density is defined:
Go ahead and calculate that and report back if you run into problems.
Generally the first integral covers all possible values. Clearly, X can't < 0, be cause that is given. Also, X cannot be >3 because then, there is no way for X+Y to be <3.
There may be exceptions, but this is the rule not the exception:
The first integral covers all possible values of one of the variables, in this case X, and the second integral covers the restricted bounds of the second variable. So,
(1)Choose a variable to restrict. Use the methods shown above to determine your bounds for that variable.
(2)Use the first integral to cover all possible values of your unrestricted variable.