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Math Help - Bivariate Variables

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    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).
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    Quote Originally Posted by maibs89 View Post
    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 |}<br />
& Y = 1 & Y = 2 & Y = 3 & Y = 4 \\ \hline<br />
X = 1 & & & & \\ \hline<br />
X = 2 & & & & \\ \hline<br />
X = 3 & & & & \\ \hline<br />
X = 4 & & & & \\ \hline<br />
\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 |}<br />
& Y = 1 & Y = 2 & Y = 3 & Y = 4 \\ \hline<br />
X = 1 & 1/8 & & & \\ \hline<br />
X = 2 & 1/24 & & & \\ \hline<br />
X = 3 & 1/24 & & & \\ \hline<br />
X = 4 & 1/24 & & & \\ \hline<br />
\end{tabular}


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