PROBABILITIES Course

Bi-variate Distribution tip

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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 $Y$.


Example

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

X\Y 1 2 3 4 5 6
1 1/36 2/36 2/36 2/36 2/36 2/36
2 0 1/36 2/36 2/36 2/36 2/36
3 0 0 1/36 2/36 2/36 2/36
4 0 0 0 1/36 2/36 2/36
5 0 0 0 0 1/36 2/36
6 0 0 0 0 0 1/36

First, you should remember that when throwing 2 dices, you got $|\Omega|=36$ results like

  • (1,1)
  • (1,2)
  • (2,1)
  • (2,2)
  • ...

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.