STAT 333 Assignment 1 Due: Thursday, May 27 at the beginning of class 1. Consider rolling a fair 6-sided die n times. Let X represent the number of faces that have NOT been rolled. a. Find the expected value of X. b. Find the variance of X. c. Describe (in words or with a graph) how the mean and variance of X behave for different values of n . Provide a brief logical explanation. (Hint: attach indicator variables to faces, not rolls. Be careful, the indicators are not indep) 2. Consider a Negative Binomial random variable Y ~ NB( r , p ). Prove that Y is a proper rv iff p > 0 by the following methods: a. Express Y as the sum of r independent Geometric random variables, and apply a result we know from class about Geometric rvs. b. Show (directly) that P(Y = ∞) = 0. c. Show (from first principles – i.e. using the pmf of a NB, or of a Geo) that E[Y] is r / p . Why does this imply Y is proper iff p > 0? 3.
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Probability theory, fair 6-sided die, Geometric random variables, brief logical explanation