I need help understanding the concepts and answering 1.1 and 1.2.

1 Practicing with MLEs Suppose you have been observing the bartender at the Mos Eisley Cantina and measuring how much he deviates from his desired volume every time he pours someone a glass of blue milk. You now have n independent and identically distributed observations X1, X2, . . . , Xn (all in milliliters). Since some deviations are above the desired volume (positive) and some are below the desired volume (negative), you have speculated that the data may follow a normal distribution with some unknown mean / and variance o? > 0. The probability density function for the normal family of distributions is given by f(x) = 1 20 for a E (-0o, 00) OV2AT As we saw in class, the maximum likelihood estimator (MLE) for u (or o) is defined as the value of u (or o) that can maximize the likelihood (i.e., joint probability density) for the given observations X1, X2, . .., Xn. Since these observations are assumed independent of each other, the likelihood can be expressed mathematically as follows: L( M, 02) = [If(Xtim, 02) = II (*) i=1 1= 1 LOV27 The task is then to find the maximizer (, o') that attains the highest value of L(M, 62). Note that L(u, 02) involves exponential functions. Therefore, it is more convenient to work on the natural logarithm of L(/, 2), which is a monotone transformation that does not change the maximizer (, 02). Specifically, the log-likelihood function is defined as I(M, 62) = In L(A, 52). 1.1 Problem Under your assumption that the iid observations X1, X2, . .., Xn come from a normal dis- tribution with unknown parameters (u, 2), derive the log-likelihood function I(/, 2) using (*). (Hint: In(ab) = In a + Inb; and In(IIt_, at) = Et, In ai.) 1.2 Problem Derive du' ", the first-order partial derivative of the log-likelihood function I(/, 02) with respect to u. Show your work