2.1 Discrete Process Generator [9 points] A factory produces two independent batches of widgets, each batch containing exactly six widgets of varying quality. During quality inspection, a sensor assigns a score from 1 to 6 to each batch (the higher the score, the higher the quality). Let Q1 = quality score of batch 1, Q2 = quality score of batch 2. Each score is uniformly distributed over the integers 1 through 6. Define the random variable X = Q1 Q2 , the absolute quality gap between the two batches. (a) 1 PT Enumerate the sample space of all possible (Q1,Q2) pairs (hint: there are 36 cases) and, for each pair, identify the value of the random variable X. (b) 1 PT As demonstrated in class, construct a table of the probability mass function (PMF) p(x) and cumulative distribution function (CDF) F(x) for 0 x 5. (c) 1 PT Compute the population mean (expected value) X = E[X]. (d) 1 PT Using an appropriate chart (see lecture 2), plot the PMF p(x) for 0 x 5. Label the axes clearly. (e) 1 PT Using an appropriate chart (see lecture 2), plot the CDF F(x) for 0