# Thread: Bivariate Variables

1. ## Bivariate Variables

A cat and dog live in a square room. The corners of the room are numbered 1, 2, 3 and 4. Each night, the cat decides at random at which corner it will sleep. The dog joins the cat in that corner with probability 1/2, but otherwise it chooses one of the remaining three cornerse at random. Denote by X the numbr of the corner the cat will sleep at, and by Y the number of the dog's corner.

Derive the pmf formula p(x,y) of the bivariate random variable (X,Y).

2. Originally Posted by maibs89
A cat and dog live in a square room. The corners of the room are numbered 1, 2, 3 and 4. Each night, the cat decides at random at which corner it will sleep. The dog joins the cat in that corner with probability 1/2, but otherwise it chooses one of the remaining three cornerse at random. Denote by X the numbr of the corner the cat will sleep at, and by Y the number of the dog's corner.

Derive the pmf formula p(x,y) of the bivariate random variable (X,Y).
Try setting up a grid and filling in the probabilities:

$\, \begin{tabular}{l | c | c | c | c |}
& Y = 1 & Y = 2 & Y = 3 & Y = 4 \\ \hline
X = 1 & & & & \\ \hline
X = 2 & & & & \\ \hline
X = 3 & & & & \\ \hline
X = 4 & & & & \\ \hline
\end{tabular}$

To get the probabilities for the first column:

Pr(Y = 1, X = 1) = Pr(Y = 1 | X = 1) Pr(X = 1) = $\left( \frac{1}{2} \right) \left( \frac{1}{4}\right) = \frac{1}{8}$.

Pr(Y = 2, X = 1) = Pr(Y = 2 | X = 1) Pr(X = 1) = $\left( {\color{red} \frac{1}{6}} \right) \left( \frac{1}{4}\right) = \frac{1}{24}$.

Pr(Y = 3, X = 1) = Pr(Y = 3 | X = 1) Pr(X = 1) = $\left( {\color{red} \frac{1}{6}} \right) \left( \frac{1}{4}\right) = \frac{1}{24}$.

Pr(Y = 4, X = 1) = Pr(Y = 4 | X = 1) Pr(X = 1) = $\left( {\color{red} \frac{1}{6}} \right) \left( \frac{1}{4}\right) = \frac{1}{24}$.

Make sure you understand where the red probabilities have come from .....

$\, \begin{tabular}{l | c | c | c | c |}
& Y = 1 & Y = 2 & Y = 3 & Y = 4 \\ \hline
X = 1 & 1/8 & & & \\ \hline
X = 2 & 1/24 & & & \\ \hline
X = 3 & 1/24 & & & \\ \hline
X = 4 & 1/24 & & & \\ \hline
\end{tabular}$

In a similar way you can fill in the other columns .....