Homework #5: More random variables

Due: May 16, 2016 at 14:00.

Read and understand the following instructions on submission of homework. If you do not follow them, you will not receive credit.

Write a plain text e-mail to me. This homework does contain special symbols, but you can copy and paste them from the assignment if you need them and don't know how to input them. (It does not use special wordprocessor features. Do not attach a wordprocessor file such as Microsoft Word or a Excel spreadsheet. I will not read them.) Give the mail the subject "01CN101 Homework #5 by <your name>" in hankaku romaji and send it to math-hw@turnbull.sk.tsukuba.ac.jp. (This subject is helpful for automatically sorting incoming mail.)

Make sure that the body of the email contains your name and student ID number.

You should automatically receive an acknowledgment of your submission by email. Keep that mail. In case I lose your mail for some reason it becomes your proof of homework submission. If you don't receive an acknowledgment, you probably submitted to the wrong address.

Problems

In these problems, unless otherwise specified, you may use the cumulative distribution, the mass function, or the density function, as is convenient.

Problem 1

Consider the state space which is the set of formations used by my daughter's dance team (each dot indicates the position of a dancer): Ω = {⚀, ⚁, ⚂, ⚃, ⚄, ⚅}, and two random variables X:Ω→R and Y:Ω→R defined by X(⚀) = 1, ..., X(⚅) = 6 and Y(⚀) = 6, ..., Y(⚅) = 1. Assume the formations are used with equal probability.

Compute the cumulative distributions of X and Y themselves, and the cumulative joint distribution of X and Y together. Conclude that X and Y are not independent. Explain why not.

Hint: in expressing the distributions, you may use any convenient table format, and you may abbreviate expressions like "X ≦ b" to just "b" in the table.

Problem 2

Construct two r.v.s X and Y such that Cov[X,Y] = 0 but X and Y are not independent. Hint: a simple way involves three values for each: -1, 0, and 1, and a condition on possible combinations so that XY = 0 for all ω. You must define the state space Ω, and prove that Cov[X,Y] = 0, and the r.v.s are not independent.

Due: May 16, 2015 at 14:00.