"When conducting a binary regression with a skewed predictor, it is often easiest to assess the need for x and log(x) by including them both in the model so that their relative contributions can be assessed directly." Show that indeed the log odds are a function of x and log(x) for the gamma distribution.

Consider linear regression where y is our label vector, X is our data matrix, w is our model weights and o² is a measure of variance Using the squared error cost function has a probabilistic interpretation as: O O O Maximising the probability of the model predicting the input data, assuming our input data follows a Normal distribution N(X; Xw, o²) Maximising the probability of the model predicting the input data given the weights N(X; wy, o²) Minimising the probability of the model predicting the labels, assuming our prediction errors follow a Normal distribution N(y; Xw, o²) Maximising the values of the weights to minimise the input data N (y; w, o²) Maximising the probability of the model predicting the labels, assuming our prediction errors follow a Normal distribution N(y; Xw, o²)

Linear regression aims to fit the parameters based on the training set T.x = 1, 2,...,m} so that the hypothesis function he (x) ...... + Onxn can better predict the output y of a new input vector x. Please derive the stochastic gradient descent update rule which can update repeatedly to minimize the least squares cost function J(0). D = {(x(i),y(¹)), i 00+ 01x₁ + 0₂x₂+... = =

In R, write a function that produces plots of statistical power versus sample size for simple linear regression. The function should be of the form LinRegPower(N,B,A,sd,nrep), where N is a vector/list of sample sizes, B is the true slope, A is the true intercept, sd is the true standard deviation of the residuals, and nrep is the number of simulation replicates. The function should conduct simulations and then produce a plot of statistical power versus the sample sizes in N for the hypothesis test of whether the slope is different than zero. B and A can be vectors/lists of equal length. In this case, the plot should have separate lines for each pair of A and B values (A[1] with B[1], A[2] with B[2], etc). The function should produce an informative error message if A and B are not the same length. It should also give an informative error message if N only has a single value. Demonstrate your function with some sample plots. Find some cases where power varies from close to zero to near…