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# HW6_08 - IE 111 Fall 2008 Homework#6 Due Wednesday October...

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IE 111 Fall 2008 Homework #6 Due Wednesday October 22 Question 1. To check on the security screeners at an airport, we put 4 fake bombs into a set of 52 baggage pieces. Suppose that the screeners actually just pick 6 pieces randomly (without replacement) to inspect. Let X be the number of our fake bombs that they find. a) According to which well known distribution is X distributed? b) Find the Probability Mass Function f(x). Remember to specify an appropriate range for x. c) Find E(X) d) Find V(X) Question 2. An urn has 6 red marbles and 14 green marbles. Careful on this one, the distribution to use keeps changing! a) If I pick 3 marbles with replacement , and let X= the number of red marbles picked, what is the P(X=1)? b) If I keep picking marbles with replacement until I get my first red marble, what is the probability I picked exactly 7 times? c) If I pick 3 marbles without replacement and let X= the number of red marbles picked, what is the P(X=1)? d) If I pick 3 marbles without replacement and let X= the number of red marbles picked, what is the expected value of X? e) Suppose I have picked 8 marbles with replacement and have not yet gotten a red marble. How many more picks do I expect will be required before my first red marble? f) Suppose I have picked 9 marbles without replacement and have not yet gotten a red marble. How many more picks (without replacement) do I expect will be required before my first red marble? This is a tricky problem! We can't use any of our favorite distributions; you have to think through it from first principles. Here's a hint to get you started. Let X = number of remaining picks. Compute P(X=1), P(X=2), P(X=3), etc. (it will help to think of the Geometric distribution, but it's not quite the same) and then compute the mean value from there. Question 3. Suppose you go shopping at a convenience store, pick your items, and then join the line for the cash register. There are 6 people in the line ahead of you, plus one person who is currently being served (there is only one clerk on duty). Having nothing better to do, you estimate that the time it takes to ring up one person’s items (measured in seconds) has a Geometric distribution with parameter p=1/30.

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HW6_08 - IE 111 Fall 2008 Homework#6 Due Wednesday October...

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