Solution: 2017 Fall Final - 21

Author: Michiel Smid

Question

You are given a fair red die and a fair blue die. You roll each die once, independently of each other. Let $(i,j)$ be the outcome, where $i$ is the result of the red die and $j$ is the result of the blue die. Define the random variables $$ X = i + j $$ and $$ Y = i - j. $$ Which of the following is true?

(a)

The random variables $X$ and $Y$ are independent.

(b)

None of the above.

(c)

The random variables $X$ and $Y$ are not independent.

Solution

Before drawing a tree, let’s think for a second

We just need to prove whether the following is true for some value $x$ and $y$: $ Pr(X=i \cap Y=j) = Pr(X=i) \cdot Pr(Y=j) $

Let’s do it for X=2 and Y=2

$ |S| = 36 $

$X=2$ for the pair $ (1,1) $

$ Pr(X=2) = \frac{1}{36} $

$Y=5$ for the pairs $ (6,1), (1,6) $

$ Pr(Y=5) = \frac{2}{36} $

Now, let’s find the case of $ X=2 \cap Y=5 $

When we look at overlapping pairs in $X=2$ and $Y=5$, we see that there is no intersection

This means that $ Pr(X=2 \cap Y=5) = 0 $

$ Pr(X=2 \cap Y=5) = Pr(X=2) \cdot Pr(Y=5) $

Since the equation is false, X and Y are not independent.