Probabilty (5 questions)

1. An urn contains 4 marbles: 2 red and 2 green. You extract them one by one without replacement. If you extract a green marble, I pay you $1; if you extract a red marble, you pay me $1.25. You can quit playing at any time. Use recursion to value this game from your perspective. Present your analysis clearly. 2. With the setup of problem 1, write down the 4 CHOSE 2= 6 color orders lexicographically, assuming you extract all four marbles. Are they all equally likely? From the results of problem 1, show the stopping rule as applied to each color order. Write down the payout for each color order and compute the expected payout. 3. Is it possible to generate random variables X and Y that are both exponentially distributed with Corr (X, Y ) = −1? If so, how. If not, why not? (Think about what you know about correlation.) [login to view URL] the Markov chain with PTM P = (pxy) is ergodic and pm(x,y) > 0 for all states x and y. If n≥m,show that pn(x,y)>0 for all states x and y. Hint: You need a path from x to y of length n. Starting from x, the first n − m steps can be anything that occurs with positive probability. 5. Suppose you wish to generate normals X, Y , and Z with the correlation matrix R but with means 0, 1, and 2, and variances 4, 16, and 25, respectively. How would you do this?