Two dice are rolled. Let $X$ and $Y$ denote, respectively, the largest and
smallest values obtained. Compute the conditional mass function of $Y$ given
$X = x$, for $x = 1, 2, · · · , 6$. Are $X$ and $Y$ independent?
The answer to the first question about the conditional pdf is given by the following table.
Now, I can understand how to manually get these values, for example, the value of $X=2$ and $Y=2$ being $1\over3$ from the fact that there are three values that involve the greatest number being $2$, which are $(1,2) , (2,1) , (2,2)$.
However, I am still confused about how to use the formula $P(Y|X) = \frac{P\{ X=x, Y=y\}}{P\{X=x\}}$ to get these values. What is the $P\{ X=x, Y=y\}$? What is the $P\{X=x\}$?
Best Answer
$P\{ X=x, Y=y\}$ is always either $1\over{36}$ if the numbers are the same, or $1\over{18}$ if they are different, or $0$ if $Y>X$.
$P\{X=x\}$ is always the number of times out of $36$ combinations that a particular number is the greatest, in my example of $X=2$ and $Y=2$, it is $3\over{36}$ as there are three choices $(1,2) , (2,1) , (2,2)$ out of $36$.