Bi-variate Distribution tip
This page may be useful if $Card(X)$ and $Card(Y)$ are small, since you would be able to use the contingency table/
tableau à double entrées/tableau de contingence.
It's a table with the columns being the values in $Y(\Omega)$ and the lines, the values in $X(\Omega)$.
The values inside the table are the probabilities of the values in the row being taken by $X$ and the value in the column to be taken by $Y$.
We are throwing 2 dices, X = "the minimum is $x_i$" and Y = "the maximum is $y_j$". We can make this a table like this
First, you should remember that when throwing 2 dices, you got $|\Omega|=36$ results like
Then for $X=1,\ Y=2$, you are looking for the probability of
- A="the minimum is 1 and the maximum is 2"
- $|A|=2$ because we have (1,2) and (2,1)
- so we have $2/36$
For $X=2,\ Y=1$, you are looking for the probability of
- A="the minimum is 2 and the maximum is 1"
- this is impossible so
For $X=1,\ Y=1$, you are looking for the probability of
- A="the minimum is 1 and the maximum is 1"
- $|A|=1$ because we have (1,1)
- so we have $1/36$
Finally, you should check and confirm that the sum of all probabilities is 1.