Answer:
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](https://tex.z-dn.net/?f=P%28X%20%3D%20k%2C%20Y%20%3D%200%29%20%20%3D%201%2F48%5C%5CP%28X%20%3D%20k%2C%20Y%20%3D%201%29%20%3D%201%2F16%5C%5CP%28X%20%3D%20k%2C%20Y%20%3D%202%29%20%3D%201%2F16%5C%5CP%28X%20%3D%20k%2C%20Y%20%3D%203%29%20%3D%201%2F48)
With k in {1,2,3,4,5,6}
Step-by-step 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](https://tex.z-dn.net/?f=P_Y%280%29%20%3D%20%7B3%20%5Cchoose%200%7D%20%2A%200.5%5E3%20%3D%201%2F8)
![P_Y(1) = {3 \choose 1} * 0.5* 0.5^2 = 3/8](https://tex.z-dn.net/?f=P_Y%281%29%20%3D%20%7B3%20%5Cchoose%201%7D%20%2A%200.5%2A%200.5%5E2%20%3D%203%2F8)
![P_Y(2) = {3 \choose 2} * 0.5^2*0.5 = 3/8](https://tex.z-dn.net/?f=P_Y%282%29%20%3D%20%7B3%20%5Cchoose%202%7D%20%2A%200.5%5E2%2A0.5%20%3D%203%2F8)
![P_Y(3) = {3 \choose 3} * 0.5^3 = 1/8](https://tex.z-dn.net/?f=P_Y%283%29%20%3D%20%7B3%20%5Cchoose%203%7D%20%2A%200.5%5E3%20%3D%201%2F8)
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](https://tex.z-dn.net/?f=P%28X%20%3D%20k%2C%20Y%20%3D%200%29%20%3D%201%2F8%20%2A%201%2F6%20%3D%201%2F48%5C%5CP%28X%20%3D%20k%2C%20Y%20%3D%201%29%20%3D%203%2F8%20%2A%201%2F6%20%3D%201%2F16%5C%5CP%28X%20%3D%20k%2C%20Y%20%3D%202%29%20%3D%203%2F8%20%2A%201%2F6%20%3D%201%2F16%5C%5CP%28X%20%3D%20k%2C%20Y%20%3D%203%29%20%3D%201%2F8%20%2A%201%2F6%20%3D%201%2F48)
For any k in {0,1,2,3,4,5,6}