Question

Suppose you roll k >= 1 fair dice. Let X be the random variable for the sum of their values, and let Y be the random variable for the number of times an odd number comes up.
Prove or disprove: X and Y are independent.

*Please use the concept of independent random variables

Answer 1

Given,

roll k>=1 fair dice

X: sum of the values from dice

Y: number of times odd number comes up

As we know, we can conclude two random variables X and Y independent if:

1. P(X|Y) = P(X), for all values of X and Y.
2. P(X∩Y) = P(X) * P(Y), for all values of X and Y

In this case, let us assume k = 2 for simplicity.

For X, the values could be [2,12] and the sample space has 36 total outcomes for 2 rolls

For Y, the values could be 0,1 or 2

As we see, to calculate

P(X=4) = 3/36 as there are only 3 possibilities (1,3), (2,2) and (3,1)

P(Y=2) = 9/36 as there are 9 possibilities (1,1), (1,3), (1,5), (3,1), (3,3), (3,5), (5,1), (5,3), (5,5)

P(X=4|Y=2) = 2/36 which are (1,3) and (3,1) which is not equal to P(X)

Hence, we can conclude that X and Y are not independent.