END OF SEMESTER PROJECT PART I 1. Toss a pair of six-sided dice, so that N = {{(1, 1), (1,2)....(6,6))}; and let X be th

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End Of Semester Project Part I 1 Toss A Pair Of Six Sided Dice So That N 1 1 1 2 6 6 And Let X Be Th 1 (87.53 KiB) Viewed 231 times

END OF SEMESTER PROJECT PART I 1. Toss a pair of six-sided dice, so that N = {{(1, 1), (1,2)....(6,6))}; and let X be the random variable on which takes the product of the roll, so that if w= (2,6), then X(W) = 12. (a) Compute u = E(X) and o? = V(X). (b) Compute the mass function fxz). and plot its graph with a computer. Now suppose E(X) and V(X) are unknown, and we want to approximate them empirically, which we do by taking random samples of dice rolls. For this, use a computer to simulate tossing a pair of dice n times for n = 50, 100, 500, 1000, 5000. Recall that the associated nath sample space is given by S?" = {((a.),..., (Org ...)) (0.,,);) € {1,...,6}"}, so that each of your simulations corresponds to an outcome on En Now let X1,..., Xo be the associated IID random variables, so that if Su-((01,b), .... (...)) then X,() = Xia, ) = 0,6,7, i.e., the product of your ith simulated toss, for i = 1,...,.. So if your 5th simulated toss is (2.5), X:(s) = 10. (c) Compute X:(s) for i = 1, 2, ... ,, plot a relative frequency diagram for n = 50, 100, 500, 1000, 5000, and compare with the graph of fx(2). Now let X be the randorn variable given by X,= X1 X2 ... | X n let S be the random variable given by S = ΣΑ, Χ. . = and let 2. be the random variable given by Vn(x,-) endom The Law of Large says that converges distribution to the constant variable di = E(X), 50 X($) should be close to y for large n.