University of California, BerkeleyCS 70FA19hw13sol.Discrete Mathematics and Probability Theory

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1 Short Answer (a) Let X be uniform on the interval [0,2], and define Y = 2X +1. Find the PDF, CDF, expectation, and variance of Y. (b) Let X and Y have joint distribution f (x,y) = (cxy 0 else +1... /4 x 2 [1,2] and y 2 [0,2] Find the constant c. Are X and Y independent? (c) Let X ⇠ Exp(3). What is the probability that X 2 [0,1]? If I define a new random variable Y = bXc, for each k 2 N, what is the probability that Y = k? Do you recognize this (discrete) distribution? (d) Let Xi ⇠ Exp(li) for i = 1,...,n be mutually independent. It is a (very nice) fact that min(X1,...,Xn) ⇠ Exp(µ). Find µ. 2 Exponential Expectation (a) Let X ⇠ Exp(l). Use induction to show that E[Xk] = k!/l k for every k 2 N. (b) For any |t| < l, compute E[etX] directly from the definition of expectation. (c) Using part (a), compute • k=0 E[kX!k]tk. Solution: (a) The base case is E[X] = 1/l, which we already know. Using integration by parts, E[Xk+1] = Z0• xk+1 · le'lx dx 3 Continuous Probability Continued For the following questions, please briefly justify your answers or show your work. (a) If X ⇠ N(0,sX2) and Y ⇠ N(0,sY2) are independent, then what is E⇥(X +Y)k⇤ for any odd k 2 N? (b) Let fµ,s(x) be the density of a N(µ,s2) random variable, and let X be distributed according to a fµ1,s1(x) + (1 ' a)fµ2,s2(x) for some a 2 [0,1]. Please compute E[X] and Var[X]. Is X normally distributed? (c) Assume Bob1,Bob2,...,Bobk each hold a fair coin whose two sides show numbers instead of heads and tails, with the numbers on Bobi’s coin being i and 'i. Each Bob tosses their coin n times and sums up the numbers he sees; let’s call this number Xi. For large n, what is the distribution of (X1 +···+Xk)/pn approximately equal to? (d) If X1,X2,... is a sequence of i.i.d. random variables of mean µ and variance s2, what is limn!• PhÂn k=1 Xsk'naµ 2 ['1,1]i for a 2 [0,1] (your answer may depend on a and F, the CDF of a N(0,1) variable)? (e) Assume we wanted to estimate the value of p by repeatedly throwing a needle of length 1 cm on a striped floor, whose stripes all run parallel at distances 1 cm from each other. Please give an estimator of p, and compute an approximate 95% confidence intervals using the central limit theorem. (Hint: You may assume that p(p ± e) ⇡ p2 for small e). Solution: 2, the reasoning is exactly as in the law of large numbers: By Chebyshev’s inequality, we have 1 ' PhÂn k=1 Xsk'naµ 2 ['1,1]i = PhÂn k=1 Xsk'naµ 62 ['1,1]i  n2a1'1 '''! n!• 0. The a = 12 case is a direct consequence of the central limit theorem, while the a < 1 2 case follows indirectly from it: PhÂn k=1 Xsk'naµ 2 ['1,1]i = PÂn k=1 Xskp'nµ 2 'n 1 21'a , n 1 21'a '' CS 70, Fall 2019, HW 13 5⇡ PN(0,1) 2 'n 1 21'a , n 1 21'a '' '''! n!• 0. (e) pˆ = 2 ·"1 n kÂ=n1I{kthneedle throw intersects a line}#'1 ,C = pˆ ± pp2p(pn'2) . Let Ik be the indicators that the k needle toss intersects one of the stripes, then d = 1 n Ân k=1 Ik converges by the law of large numbers and what we have learnt about Buffon’s needle to p2 , and so pˆ = 2 d is a reasonable estimator of p. Moreover, P+ + + +d2 'p+ + + +  e' = Pp 'e  d2  p +e' = Pp +2 e  d  p '2 e ' = P2ps n ✓p +1 e ' p1 ◆  pn(ds'2/p)  2ps n ✓p '1 e ' p1 ◆' ⇡ P'2sp pn2e  N(0,1)  2sp pn2e ' =! 0.95, where s = pVar[Ik] = qp2 ,1' p2- = p2(pp'2). We know that the 95% confidence interval for a normal distributions is achieved roughtly at ±2, and so solving 2pne sp2 = 2 gives e = pp2p(p n'2). 4 Bu↵on’s Needle on a Grid In this problem, we will consider Buffon’s Needle, but with a catch. We now drop a needle at random onto a large grid, and example of which is shown below. The length of the needle is 1, and the space between the [Show More]

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