Marginal pdf

[maz1987]

Pg 9 of Chapter 6 the notes gives an example of a joint pdf:

f(x,y) = 0.5 if x>0, y>0, x+y<2
.........= 0 otherwise

It uses 2-x as the upper bound of the integral used in order to obtain the marginal pdf of X. I understand this.

However I modified it slightly to give an example of a lower bound involving the variable:

f(x,y) = 0.5 if x> -2, y>0, x+y>2
.........= 0 otherwise.

Would I integrate between 2-x and infinity to obtain the marginal distribution of X in this case?

And, more generally, if we have one constraint that involves both variables, do we put the variable that we're removing as the subject, and then use that as the intuitive boundary (e.g. if f(x,y)< something it's the upper boundary, assuming y has a positive sign, and if f(x,y)> something it's the lower boundary).

Thanks

 

[John Lee]

The general principal of to get f(x) integrate over y and use limits for y in terms of x is sound.

Your example doesn't work as the area is infinite - so the double integral won't sum to 1 and more importantly your lower limit isn't quite the lower limit for all the values of y as y > 0 comes into play when x > 2.