Random numbers and sampling from uniform distrubution

Question

U1 ~ Uniform[0,1] and U2 ~ Uniform[2, 3]. In which region of the U1-U2 plane is the joint pdf of U1,U2 nonzero?

I have not understood the meaning of U1-U2 plane here.. Can anyone please explain how do I go about solving this question?

Answer 1

When you have two random variables, you can imagine each outcome as a point in the plane, since each outcome consists of a value for $U_1$ and a value for $U_2.$ For a continuous distribution, there will be a density $f_{U_1,U_2}(u_1,u_2)$ for the outcome occurring near every point $(u_1,u_2)$ in the plane, such that the probability of the outcome lying in a certain region is the integral of the density function over that region. This is what a joint PDF is.

The region of the plane where the PDF is nonzero is called the support, so this problem is asking you for the support of the PDF. Here, as you've indicated in the comments, the variables are independent, so the joint PDF is just the product of the marginal PDFs. Since they're uniform, we can write the marginal PDFs $$f_{U_1}(u_1) = 1_{[0,1]}(u_1) \\f_{U_2}(u_2) = \frac{1}{3-2}1_{[2,3]}(u_2)= 1_{[2,3]}(u_2),$$ where $1_A(u)$ is the indicator function that is $1$ when $u\in A$ and zero otherwise. By independence the joint is $$ f_{U_1,U_2}(u_1,u_2)=1_{[0,1]}(u_1)1_{[2,3]}(u_2)$$ so your task is to write down the region of the $u_1-u_2$ plane where this function is nonzero.

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A couple of comments. First, I've used a lot of notation here, but really, at least in the independent case, this is pretty much common sense once you understand everything. It's basically equivalent to

$u_1$ can be any number between zero and one, and $u_2$ can be any number between $2$ and $3.$ What region of the plane can $(u_1,u_2)$ lie in?

but I'll stress this is only in the case of independence.

Second, much less important comment for completeness's sake. This is not actually all that well-posed since you can always change the PDF on a set of measure zero (including setting the PDF to zero there) and still have the same distribution. So 'places where the PDF is zero' is not quite the same thing as 'support'. This is really a minor technical point you can probably ignore for now.