[Solved] CSE312-Assignment 8

1. Maestro Lazy decides to assign final grades in CSE 312 by ignoring all the work thestudents have done and instead using the following probabilistic method: each student independently will be assigned an A with probability , a B with probability 4, a C with probability , and an F with probability . When the quarter is over, you discover that only 5 students got an A, 25 got a B, 100 got a C, and 90 got an F. Find the maximum likelihood estimator for the parameter that Maestro Lazy used. Give an exact answer as a simplified fraction.
2. Maestro Lazy decides to assign final grades in CSE 312 by using a more sophisticatedprobabilistic method: each student independently will be assigned an A with probability ₁, a B with probability _{2 1}, a C with probability , and an F with probability

. When the quarter is over, you discover that only 5 students got an A, 25 got a B, 100 got a C, and 90 got an F. Find the maximum likelihood estimators for the parameters ₁ and ₂ that Maestro Lazy used. Give exact answers as simplified fractions. For this problem, you do not have to demonstrate that your estimators are local maxima, which turns out to be more challenging than just computing second derivatives.

3. (a) Let x₁,x₂,,x_n be independent samples from a geometric distribution with unknown parameter p. What is the maximum likelihood estimator for p?

(b) If the samples from the geometric distribution are 5, 4, 3, 2, 9, 5, 3, 9, what is the maximum likelihood estimator for p? Give an exact answer as a simplified fraction.

4. Let x₁,x₂,,x_n be independent samples from an exponential distribution with unknown parameter . What is the maximum likelihood estimator for ? (If you do Exercises 3a and 4 correctly, you should see a very natural relationship between the estimators.)
5. (a) Suppose x₁,x₂,,x_n are the numbers of independent requests arriving at a web server during each of n successive minutes. Under the assumption that these numbers are independent samples from a Poisson distribution with unknown rate per minute, what is the maximum likelihood estimator for ? (If you do Exercises 4 and 5a correctly, you should see a very natural relationship between the estimators.)
   • Is your estimator unbiased? Justify your answer.
6. I have played 12273 games against the devilish Schnapsen program Doktor Schnaps and have won 5926 of them. Lets suppose that each games outcome X_i is Ber(p), where p is my probability of winning a single game, and that the outcomes are all independent. The goal of this exercise is to estimate my long-term winning probability p against Doktor Schnaps. Since Doktor Schnaps and I are nearly evenly matched, you can use the estimate Var(X_i) = 0.25 (which would be exact if). Give all your answers to 4 significant digits.
   • What is the maximum likelihood estimate p of p for the 12273 games I have played? You do not need to rederive the formula for this estimator, since we did it in lecture. Just substitute the numbers into the formula to arrive at a numerical estimate.
   • Approximate the 99% confidence interval for p. Give your answer as the least value of such that P(p p p + ) 0. (For consistency for the graders sake, when you use the standard normal distribution table, choose the entry that barely makes this inequality true.) Once you have provided , also specify the confidence interval [p ,p+ ] to 4 significant digits.

For your interest, Doktor Schnaps actually makes use of these results: the players score in the last column of the leaderboard at http://schnapsen.realtype.at/index.php? page=gesamtwertung is the lower boundary p of the 99% confidence interval. This choice prevents a player who has only played a few lucky games from being ranked too high, since decreases as the number of games played increases. You can see examples of how this affects ranking in the leaderboard table, where the next-to-last column is p but the table is sorted on the last column p .

7. (a) Let x₁,x₂,,x_n be independent samples from the continuous uniform distribution on [0,], where is the unknown parameter. What is the maximum likelihood estimator for ?

Hint: It might help to roughly sketch out the shape of the likelihood function as a function of , that is, on the horizontal axis and likelihood L on the vertical axis. Start your sketch by asking what happens to L as goes to infinity. Then figure out the shape as you get closer to = 0. From your sketch, at what value of do you maximize L?

• Does your answer to part (a) seem as though it would produce the most accurateestimator for ? Explain.
• In the remaining parts of this exercise, you will quantify your answer to part (b) by computing the bias of the estimator . Begin by computing the cumulative distribution function for the random variable . Recall that this is simply the function F(x) = P( < x). Focus first on the interval 0 x , but when youre done with that, dont forget to also define F(x) on the rest of the real numbers.
• From your answer to part (c), compute the probability density function f(x) of .
• From your answer to part (d), compute E[]. Is an unbiased estimator of ? Was your intuition in part (b) consistent with your answer to this part?
• Starting from the value of E[] you computed in part (e), determine an unbiased estimator of and show that it is unbiased. Although there are other unbiased estimators of , you are to derive yours by modifying your answer from part (e).