Probability joint distribution on probability mass function Two dice are tossed. Let X be the smaller number of points. Let Y be the larger number
of points. If both dice show the same number, say, z points, then X = Y = z.
(a) Find the joint probability mass function of (X, Y ).
(b) Are X and Y independent? Explain.
(c) Find the probability mass function of X.
(d) If X = 2, what is the probability that Y = 5?
 A: Let $X_1$ denote the value of the first dice, $X_2$ denote the value of the second dice. Then $X=\min(X_1,X_2),Y=\max(X_1,X_2)$. Then for $x\neq y$,
$$P(X=x,Y=y)=P(X_1=x,X_2=y)+P(X_1=y,X_2=x)=2P(X_1=x)P(X_2=y)$$
and for $x=y$, 
$$P(X=x,Y=x)=P(X_1=x,X_2=x)=P(X_1=x)P(X_2=x)$$
noting $X_1$, $X_2$ are independent. You can easily find out the joint pmf from this.
If the joint could be written as product of two functions, one depending only on $x$ and the other only on $y$, then $X$ and $Y$ are independent, otherwise not.
Once you get the joint pmf, take it's sum over possible values of $Y$ to get the pmf of $X$. 
A: The following sheet shows the joint distribution of the min and the max of two rolls:

The min and the max are not independent:
For min=1 and max =3, the probability that the min is 1 is the sum of the numbers in the blue frame, $\approx 0.305$; the probability that the max is 3 is the sum of the numbers in the green frame, $\approx 0.138$; the product of these two numbers is $\approx 0.424$ and the probability that the min is 1 and the max is 3 is  $\approx 0.555$.
The probability that the max is $5$ is the sum of the numbers in the red frame.
