**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.

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.

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.**