Question: Problem 5 Let X1, ..., Xn be i.i.d random variables with cumulative distribution function F. In other words, Vi , P(Xi X(n) the order statistics

Problem 5 Let X1, ..., Xn be i.i.d random

Problem 5 Let X1, ..., Xn be i.i.d random variables with cumulative distribution function F. In other words, Vi , P(Xi X(n) the order statistics of the sample. For simplicity, we assume that F is continuous. 1. Call Y{(x) the random variable (1 if X, SI Yi(x) = 0 otherwise. Show that {X(j) sx} if and only if m=1 Ym(x) > j . 2. What is the distribution of Sn(x) = {m=1 Ym (2), if x is fixed? (I.e to what family does it belong, and with what parameters?) 3. What is the cdf of max X;? What is its density? 4. What is the cdf of min X;? What is its density? 5. What is the cdf of X(j) for any j? What is its density? 6. What is E (x(j)) if Xi's are uniform on [0,1]? 7. Explain how you could check this last result with R/with a small program. (You do not have to do it - just explain what you would do.)

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