Question

Suppose X is uniform of (0,1) and Y=X. Find the joint distribution function of X and Y.

Answer 1

forget Y for a second and ask yourself: what is the marginal PDF of X?

Answer 2

Is it just 1?

Answer 3

yes, and if y=x then the joint must be... ?
(important to keep in mind the conditions for which this is valid)

Answer 4

So y would also be 1, making the joint 1 as well right?

Answer 5

right, but you should also right the interval and condition for which this is true

Answer 6

Which is same as the original, (0,1)?

Answer 7

with the additional condition that y=x though\[f_{X,Y}(x,y)=\begin{cases}1,&&if~~x\in(0,1)~~and~~y=x\\0,&&otherwise\end{cases}\]

Answer 8

Okay that makes sense. Thanks again!