Advanced Statistical Inference I
Homework 1: Probability Theory Due Date: October 7th 1. (Detect mixture distribution) Exercise 1.6.
2. (Countable additivity and Kolmogorov’s Axiom) Exercise 1.12 and Exercise 1.35.
3. (Information and Conditioning) Exercise 1.32 and Exercise 1.37.
4. (Is there a cheating?) Exercise 1.22.
5. (Sampling and central limit theorem) (a) Exercise 1.28.
(b) Exercise 1.31.
6. (Model time series and etc) (a) Exercise 1.38.
(b) US weather service has a record for each January from 1948 to 1983 at the station of Snoqualmie falls. According to the definition of raining day, there are 325 sunny days and 791 rainy days. Someone proposes a model on pi, which is the probability of a particular day, by a coin-tossing model in which pi = p and the outcomes are independent. Under this modeling, derive the probability of getting 325 sunny days in 1116 days.
(c) (Continuation of (b)) What kind of p will maximize the occurring of such an event?
(Here the event refers to 325 sunny days in 1116 days.)
(d) Another person proposes an alternative model in which pi will depend on pi−1. In particular,
pi =
( pw if the (i − 1)th day is rainning pd if the (i − 1)th day is not rainning The available data for this modeling is as follows:
Today is sunny Today is rainy Total
Yesterday is sunny 189 123 309
Yesterday is rainy 128 643 771
Total 314 766 1080
Please write down the probability of getting 791 rainy days in 1116 days.
7. (What is the meaning of random assignment? Think of lottery.) (a) Exercise 1.20 and Exercise 1.46.
(b) What is the probability distribution of X1?
(c) If someone does this experiment once, you are asked to guess the outcome of X1 defined in Exercise 1.46. Let Yj denote the rule that someone will make a guess j. If the guess matches the outcome of X1, a prize of 100 dollars will be given.
Otherwise, there is no reward. Determine the expected return of rule Yj and decide which Yj gives the highest return.
8. (Comparison of two random variables) (a) Exercise 1.49.
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(b) Let X and Y be two random variables defined as in Example 1.5.4 with pX and pY, respectively. Here pX and pY refer to the probability of a head on any given toss for those two coins, respectively. When pX > pY, is FY is stochastically greater than FX? Give reason to support your conclusion.
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