S21 1201 HW3 Solutions

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Ohio State University *

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Statistics

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Feb 20, 2024

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Topics covered 2.4 Conditional probability 2.5 Independence 3.1 Introduction to Random Variables 3.2 Probability distributions 4.1 Probability density functions 4.2 Cumulative distribution functions. 1. 3.1 Problem 10 : The number of pumps in use at a 6 pump and a 4 pump station will be observed tomorrow at noon. Give the possible values for each random variable a. T, the total pumps in use T = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} b. X the difference in the number of pumps in station 1 and station 2. X = {-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6} c. U the maximum number of pumps in use at either station U = {0, 1, 2, 3, 4, 5, 6} d. Z, the number of stations having exactly two pumps in use. Z = {0, 1, 2} 2. 3.2 Problem 19 : The textbook is posted P(Y=0) = P(Both arrive on Wed) = .3 * .3 = .09 P(Y=1) = P(W,Th) OR (Th,W) OR P(Th,Th) = .3*.4 + .4*.3 + .4*.4 = .4 P(Y=2) = (W,F) OR (Th,F) OR (F,W) OR (F,Th) OR (F,F) = .32 P(Y=3) = 1 - (.09+.4+.32) = .19

3. You have three uniform random variables U(0,1), X,Y,Z. Either using calculus or Monte Carlo simulations (i.e. a ton of random samples on the computer). This is much easier with simulations than with calculus a. What is the probability density function of X + Y + Z? b. What is the cumulative probability function of X + Y + Z? Graph it. 4. An urn contains n balls numbered 1,2, ... ,n. We select at random r balls. (a) with replacement, (b) without replacement. What is the probability that the largest selected number is m? a) (m/n) r - (m-1 / n) r b) [ (M-1)C(R-1) ] / [ (N)C(R) ]

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