Can you please solve part b:

Logistic regression estimating equations 2. (6 points) When 3;,- is a binary 0-1 variable, it is awkward to model it as a linear function of xi, since it is bounded: any line we choose, as long as it has nonzero slope, will eventually leave the interval [0,1]. A more natural choice is to estimate some linear function and then pass it through a. nonlineolty. For various reasons, the most popular choice of nonlinearity is the logistic function ez 1 g): 1+ez : 1+e'z' This is also sometimes called a sigmoid function because if the plot looks a bit like the letter S. A (simple) logistic regression is a regression of a binary response :9 on a single variable 1;, where we estimate 3:} = f (a + (3:12), for f as dened above. Note that because 3,; is always 0 or 1, it may seem nonsensical to predict it to be, e.g., 0.7. It does, however, make sense to predict that it has a 70% probability of being 1 and a 30% probability of being zero. We could try to use the squared error loss or absolute loss to estimate a simple logistic regression, but it is more natural to use the log-likelihood, i.e. the probability that 311,. . . ,yn would come out exactly the same way they did if we were to collect an new data points. If y1,. . . ,yn were independent, and each 3;,- had a probability exactly 3},- of turning out to be 1, then the corresponding loglikelihood loss (or log loss for short) is L( A) log(j') if y = 1 T.T : J' J 10s(1 :9) if y : 0 (b) (3 points) Give an expression for the empirical risk with respect to the logloss and take derivatives to Obtain estimating equations for E}, and (3. Show that the estimating equations are exactly the same as the estimating equations for linear regression: except for the one difference that the form of 3,7!- is f (a + bx) instead of a+b$