Question

g Consider the experiment of a single roll of an honest die and a single toss of 3 fair coins. Let X be the value on the die and let Y be the number of heads obtained on the coins. Find the joint probability function of X and Y

Answer 1

The probability function of X and Y is

`P(X = k, Y = 0) = 1/48\\P(X = k, Y = 1) = 1/16\\P(X = k, Y = 2) = 1/16\\P(X = k, Y = 3) = 1/48`

With k in {1,2,3,4,5,6}

Explanation:

We can naturally assume that X and Y are independent. Because of that, P(X=a, Y=b) = P(X=a) * P(Y=b) for any a, b.

Note that, since the die is honest, then P(X=k) = 1/6 for any k in {1,2,3,4,5,6}. We can conclude as a consequence that P(X=k, Y=l) = P(Y=l)/6 for any k in {1,2,3,4,5,6}.

Y has a binomial distribution, with parameters n = 3, p = 1/2. Y has range {0,1,2,3}. Lets compute the probability mass function of Y:

`P_Y(0) = {3 \choose 0} * 0.5^3 = 1/8`

`P_Y(1) = {3 \choose 1} * 0.5* 0.5^2 = 3/8`

`P_Y(2) = {3 \choose 2} * 0.5^2*0.5 = 3/8`

`P_Y(3) = {3 \choose 3} * 0.5^3 = 1/8`

Thus, we can conclude that the joint probability function is given by the following formula

`P(X = k, Y = 0) = 1/8 * 1/6 = 1/48\\P(X = k, Y = 1) = 3/8 * 1/6 = 1/16\\P(X = k, Y = 2) = 3/8 * 1/6 = 1/16\\P(X = k, Y = 3) = 1/8 * 1/6 = 1/48`

For any k in {0,1,2,3,4,5,6}