Final answer:
To calculate the pmf of U = XY + 1 with X following Binomial(1,1/3) and Y following Binomial(2,1/2), one must find all possible values of XY, add 1, and then sum the probabilities of the combinations resulting in the same U value, using the binomial probabilities formula.
Step-by-step explanation:
To calculate the probability mass function (pmf) of U, where U = XY + 1, we first need to understand the distributions of X and Y. Here, X follows a Binomial(1,1/3) distribution, and Y follows a Binomial(2,1/2) distribution. Since the binomial distribution only allows for integer values as outcomes, we can list the possible values of X and Y and then calculate the probabilities of U based on these.
The random variable X can take values 0 (failure) or 1 (success), with probabilities q = 2/3 and p = 1/3 respectively. The random variable Y can take 0, 1, or 2 successes in its trials, with the corresponding binomial probabilities. By multiplying the possible results of X and Y and then adding 1, we obtain the possible values of U. Afterward, by summing the probabilities of all combinations leading to the same U, we obtain the pmf for U.
It's important to keep in mind that when dealing with discrete random variables like in this question, we need to use precise methods to calculate the pmf, such as the formula for binomial probabilities, which is P(X = k) = (n choose k) p^k (1-p)^(n-k).