Homework #5: More random variables

Due: May 16, 2016 at 14:00.

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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.