2. A discrete random variable X has the following pmf: m 1 2 3 Mar) .92 260-9) (19)? A random sample of size 'n. = 20 produced the following observations: 1, 1, 2, 3, 1, 2, 1, 3, 2, 2, 2, 1, 3, 1, 3, 1, 1, 2, 1, 2. For each of the following quantities, derive a general formula and, where applicable, calculate it using the given data. (a) i. Find IE(X) and var(X). ii. Find the method of moments estimator and estimate of 9. iii. Find the standard error of this estimate. (b) Let H, '2 and Hg denote the sample frequencies of 1, 2 and 3, respectively. i. Fin the likelihood function in terms of F1, F2 and F3. ii. Find that the maximum likelihood estimator and estimate of 9. iii. Find the variance of this estimator. (Hint: write the estimator in terms of the sample mean.)