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HOMEWORK 4 PSTAT

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Numerical answers in random order: 11=243; 27e−3 27e−3 + 25e−5 ; SX = f−2; −1; 0; 1; 2; 4g; 0:0127; 11−−13 4ee−−33 ; 0:648; i −3 1(0:6)4(0:4)i−4; (17=35)n−1(18=35); 4=13; ... SX = f0; 1; 2; 3; 3:5g; k pX(k) -2 6/91 -1 8/91 0 1/91 1 32/91 2 16/91 4 28/91 ; k pX(k) 0 0.28 500 0.27 1000 0.315 1500 0.09 2000 0.045 ; k pX(k) 0 1/2 1 1/10 2 1/5 3 1/10 3.5 1/10 1. Suppose we draw 3 marbles from a jar containing 5 red and 7 blue marbles. At each stage, a ball is drawn and its color noted. Then the marble is returned to the jar along with another marble of the same color. Given that the first marble is red, what is the probability that the next two are blue? Solution: Ri = fith draw is redg; Bi = fith draw is blueg P (B2 \ B3jR1) = P (B2jR1)P (B3jR1 \ B2) = 7 13 · 8 14 = 4 13 2. (adapted from Ross, 4.1) Two candies are chosen randomly from a bag of Valentine’s Day candy containing 8 Snickers (S), 4 Peanut Butter Cups (P), and 2 Baby Ruths (B). Suppose that we win $2 for each S and we lose $1 for each P selected. Let X denote our winnings. (a) What is the state space of X? Solution: SX = f−2; −1; 0; 1; 2; 4g (b) What is the probability mass function of X? Solution: k pX(k) -2 6/91 -1 8/91 0 1/91 1 32/91 2 16/91 4 28/913. (Ross, 4.19) Suppose that the distribution function of X is given by FX(b) = 8>>>>>>>>><>>>>>>>>>: 0 b < 0 1=2 0 ≤ b < 1 3=5 1 ≤ b < 2 4=5 2 ≤ b < 3 9=10 3 ≤ b < 3:5 1 b ≥ 3:5 (a) What is the state space of X? Solution: SX = f0; 1; 2; 3; 3:5g (b) Calculate the probability mass function of X. Solution: k pX(k) 0 1/2 1 1/10 2 1/5 3 1/10 3.5 1/10 4. Naruto is taking an important multiple choice ninja exam that is 5 questions long. Each question has 3 possible answers. What is the probability that Naruto will get 4 or more correct answers just by guessing? (If X is the (random variable giving the) number of correct answers Naruto gets while guessing, then convince yourself that X = Bin( d n; p) for n trials and a good choice of p.) Solution: Let X be the number of questions that Naruto answers correctly. Then, X = Bin(5; 1=3). P(X = 4 [ X = 5) = 5 4 134 2 31 + 5 5 1 35 2 30 = 243 11 5. (adapted from Ross, 4.60) Suppose that the number of times that Vampire Bill gets attacked by Hep V in a given year is a Poisson random variable with parameter λ = 5. Now, he discovered a new drug (based on large quantities of vitamin C) that has just been marketed to protect from Hep V. It claims to reduce the Poisson parameter to λ = 3 for 75% of the population. For the rest of the population, the drug has no appreciable effect on Hep V. If Bill tries the drug for a year and only had 2 Hep V attacks in that time, how likely is it that the drug was beneficial for him? [Show More]

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