IE111_F09_HW04_soln

IE111_F09_HW04_soln - IE 111 Fall Semester 2009 Homework #4...

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IE 111 Fall Semester 2009 Homework #4 Due Friday, 10/2 Question 1. X is a Binomial RV with parameters N=400 and p=0.5. What is the formula for Pr{X=200}? Pr{X=200} = 400 C 200 *p 200 (1-p) 200 Using both a calculator and Excel, try to calculate the two components of Pr{X=200} namely 400 C 200 and 0.5 200 . What goes wrong? Compare to the results you get using the EXCEL function BINOMDIST. Actually, 0.5 400 is so small that many calculators think it is 0. Similarly, 400 C 200 is so large (above 10^100) that many calculators get an overflow error. Actual result is with excel: Pr{X=200} = 400 C 200 *p 200 *(1-p) 200 = 400 C 200 *0.5 200 *(1-0.5) 200 = 0.03986 Question 2. A truck is carrying 100,000 apples of which 5% have worms. I inspect 100 apples. a) What is the probability that I find five or fewer bad apples? Let X = the number of bad apples out of the 100 sampled. X is hypergeometric since we sample apples without replacement. Since we sample without replacement, the probabilities change with each apple selected. However, due to the large number of apples in the shipment, the probabilities change very little. Thus we can approximate the Hypergeometric distribution with a binomial with N=100 and p=0.05. P(X 5) = P(X=0) + P(X=1) +. ...+P(X=5) = 0.615999 A very similar number is obtained from the hypergeometric distribution. b) A new truck arrives, only this time I don’t know the percentage that have worms. I decide to reject the truckload of apples if I find six or more bad apples in a sample of 100. What is P(I reject the truckload |2% are bad) What is P(I reject the truckload | 8% are bad) for p = 0.02, P(X 6) = 1 - P(X 5) = 1 - [P(X=0) + P(X=1) +. ...+P(X=5)] = 1 - 0.984516 = 0.015484 for p = 0.08, P(X 6) = 1 - P(X 5) = 1 - [P(X=0) + P(X=1) +. ...+P(X=5)] = 1 - 0.179876= 0.820124 c) Consider P(I reject the truckload | X% are bad). Plot a curve with P(I reject the truckload | X% are bad) on the y-axis and X% on the x-axis. Excel would come in handy here. Handdrawn figure are also fine. Method 1: Use EXCEL to plot 1
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Method 2: Use Matlab to plot 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 X P P ( I r e j e c t t h e t r u c k l o a d | X % a r e b a d ) You can write the codes in M.file: x=linspace(0,100,101); M=100000;K=x*1000; y=1-hygecdf(5,M,K,100); plot(x,y,'.') grid; axis([0,105,0,1.05]) 2
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Question 3. An experiment consists of rolling 5 dice. Define random variable X to be the maximum of the five numbers showing. We seek to find the distribution of X. Since we cannot figure out how to answer this question analytically, we decide to use Monte Carlo simulation in EXCEL to approximate the distribution of X. Use EXCEL to simulate
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This note was uploaded on 02/21/2010 for the course IE 111 taught by Professor Storer during the Spring '07 term at Lehigh University .

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IE111_F09_HW04_soln - IE 111 Fall Semester 2009 Homework #4...

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