HW4 (Due 10/17)

Using the KKT conditions, please try to show that the mode of \(Dir(\alpha_1,\cdots,\alpha_n)\) is \(\frac{\alpha_i-1}{\alpha_1+\cdots+\alpha_n-n}\)

Let \(X_1\) and \(X_2\) be two discrete random variables whose outcomes do not overlap. And let \(X=X_1\) with prob \(p\) and \(X=X_2\) with prob \((1-p)\). Show that \(H(X) = p H(X_1) + (1-p) H(X_2) + H(p)\)

A group of students have the average height of 170 cm and the standard deviation of 15 cm. How many bits on average are approximately needed to store the height of a student with the precision of 0.1 cm if the height is normally distributed?