Final answer:
The PMF, or Probability Mass Function, for the sum of two rolls of a fair 4-sided die can be calculated by considering all possible outcomes. The probability of obtaining a sum greater than 5, given that the sum is at least 3, can be found by calculating the conditional probability. The expected value of the sum, the expected value of the sum squared, and the variance of the sum can also be calculated using the probabilities obtained from the PMF.
Step-by-step explanation:
(a) The PMF, or Probability Mass Function, for the sum of the two rolls can be found by considering all possible outcomes. Since we are rolling a fair 4-sided die twice, there are a total of 4 * 4 = 16 possible outcomes. Let's denote the sum of the two rolls as Y. We can calculate the probability for each value of Y using the PMF formula:
P(Y=y) = number of successful outcomes / total possible outcomes
For example, if y=2, there is only one successful outcome (both rolls are 1), so P(Y=2) = 1 / 16. Similarly, if y=5, there are two successful outcomes (rolls of 2 and 3, or rolls of 3 and 2), so P(Y=5) = 2 / 16. Proceeding in this manner, we can calculate the PMF for all values of Y.
(b) To find P[Y>5], we need to sum up the probabilities for all values of Y greater than 5. In this case, the sum will be P(Y=6) + P(Y=7) + P(Y=8). Using the results from part (a), we can plug in the values and calculate the probability.
(c) P[Y>5|Y>=3] represents the probability of getting a sum greater than 5, given that the sum is at least 3. In mathematical notation, it can be calculated as P(Y>5 and Y>=3) / P(Y>=3). To find P(Y>5 and Y>=3), we need to consider the successful outcomes where the sum is both greater than 5 and at least 3. Using the results from part (a), we can calculate this probability. Then, to find P(Y>=3), we sum up the probabilities for all values of Y greater than or equal to 3.
(d) The expected value of Y, denoted by E[Y], represents the average value we would expect to obtain if we repeated the experiment multiple times. It can be calculated as the sum of the product of each possible value of Y and its corresponding probability. Using the results from part (a), we can calculate this expected value.
(e) The expected value of Y squared, denoted by E[Y^2], represents the average value of Y squared. It can be calculated as the sum of the product of each possible value of Y squared and its corresponding probability. Using the results from part (a), we can calculate this expected value.
(f) The variance of Y, denoted by var(Y), represents the measure of how spread out the values of Y are from the average. It can be calculated as the sum of the product of each possible value of Y minus the expected value of Y squared, and its corresponding probability. Using the results from parts (a) and (d), we can calculate this variance.