I was hoping somebody could explain to me how you go about determining the domain for the marginal distribution, the marginal expectations, and the marginal variances. Here is the problem:
Let $f(x,y)= \left\lbrace \begin{matrix} 2 & 0\le y\le x \le 1 \\ 0 & otherwise \end{matrix} \right. $
Find $μ_x, \sigma^2_x,$ and $f(x)$.
I don't need somebody to solve this for me, as I already have the solution. However, I do not really understand how the domains of these three functions work. Of a similar vein, I do not understand which limits of integration to use at each step throughout the problem. Specifically, I'm trying to figure out:
- When solving the integral for $f(x)=\int f(x,y)dy$, it is intuitive to me that you would integrate between $0$ and $x$. However, I do not understand why the domain for $f(x)$ is $0\le x \le 1$.
- After you solve for the expected value of $x$, why is its domain $0\le x \le 1$?
- What would the domain be for the variance of $x$? Is there an intuitive way to understand the domain for a marginal variance where the original function's domain for $x$ depends on $y$ (and vice versa), such as this one does?
I would greatly appreciate anybody who could clear this up for me. I have been unable to find a good explanation for these questions in my textbook. Thanks in advance!