Let the random variable \(X\) represent the product of the number of dots facing up on each die after a pair of fair dice is rolled. Let \(Y\) represent the sum of the number of dots facing up on the pair of dice.

a. Define a probability model \((R(X), f(x))\) for the random variable \(X\).

b. What is the probability that \(X \geq 16\) ?

c. Define a probability model \((R(X, Y), f(x, y))\) for the random vector \((X, Y)\).

d. What is the probability that \(X \geq 16\) and \(Y \geq 8\) ?

e. Are \(X\) and \(Y\) independent random variables?

f. Define the conditional PDF of \(X\) given that \(Y=7\).

g. What is the probability that \(X \geq 10\) given that \(Y=7\) ?