There are three different incomes, x, and their proportions, f (x).
$ 10,000 0.40
$ 30,000 0.40
$ 50,000 0.20
How do I calculate the joint distribution for X1 and X2, which are a random sample of two incomes?
I'm new to this, so any help is appreciated. I feel like the proportions are important here, but I don't think I'd multiply the income by the proportion.
You will have to make some assumptions. Probably you are expected to assume that the random variables $X_1$ and $X_2$ are independent. Under that assumption, which you should state explicitly, we have $\Pr(X_1=a \land X_2=b)=\Pr(X_1=a)\Pr(X_2=b)$.
Now we just need to compute. For example, $\Pr(X_1=10000\cap X_2=10000)=(0.4)(0.4)$, $\Pr(X_1=10000\cap X_2=30000)=(0.4)(0.4)$, and $\Pr(X_1=10000\cap X_2=50000)=(0.4)(0.2)$. There are $6$ other entries.
Your notation might be different. You may be expected to write $f_{X_1,X_2}(10000,10000)=(0.4)(0.4)$, with $8$ other similar expressions.