HW2 (Due 10/3)

  1. (5 points) With the setup as on Slide 43 of the lecture slides, recompute the a) ML, b) MAP, and c) Bayesian estimate of the probability of the fourth toss being head when we have two heads and then a tail.

  2. (5 points) Repeat Q1 but assume that the prior is \(Beta(a=0.5,b=0.5)\) instead (slides 85-86 may help). Note that when \(a\) or \(b\) is less than 1, the beta distribution is multimodal. The mode is at one of the two extremes (0 or 1) instead of \(\frac{a-1}{a+b-2}\). (Thanks Trey for catching my earlier mistake of stating the above distribution to be skew rather than multimodal).

  3. (10 points) Continue with tropical island question in HW1. Let's denote \(W_i\) as the weather condition (either sunny or rainy) and \(T_i\) as the temperature on day \(i\).

    1. What is \(H(T_0)\)?

    2. What is \(H(T_0|W_0)\)?

    3. What is \(H(T_1|W_0)\)?

    4. What is \(I(T_1;T_0|W_1)\)?

  4. (5 points) Consider sequences of \(10,000\) coin-flips with the probability of head equal to \(0.7\). Show that a sequence with \(7,000\) heads and \(3,000\) tails is typical.