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Lecture_1 Exercises

# Lecture_1 Exercises - X c Find an expression for 1 P X = d...

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Lecture 1 The goal of this lecture is to review some basic ideas from STAT 230 including basic probability models some common discrete and continuous random variables Exercises 1. Suppose a fair die is rolled repeatedly and, on each roll, you note whether or not a six occurs. a) Explain why this is a sequence of Bernoulli trials. b) If X is the number of sixes in the first 10 rolls, find ( 2) P X = . c) What is ( 2) P X ! ? d) Let Y be the number of rolls until the first six occurs. Find ( 3) P Y = . e) Find the probability that Y is odd. f) Let Z be the number of rolls until the third six occurs. What are the possible values of Z ? g) Find ( 12) P Z = 2. Suppose a carton contains 1000 electrical fasteners of which 5 are defective. You select a sample of 20 fasteners at random without replacement (don’t ask how this might be done!). Let X be the number of defective fasteners in the sample. a) What type of random variable is X ? b) What are the possible values (the support) of
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Unformatted text preview: X ? c) Find an expression for ( 1) P X = . d) Explain why and how to use a binomial probability to approximate ( 1) P X = . 3. Suppose, in a Poisson process, there are 3 events on average per hour. a) What is the probability of exactly 2 events in the first hour? b) What is the probability of at least 2 events in the second hour? c) What is the probability of exactly 2 events in the first hour and at least 2 events in the second hour? d) What is the probability of exactly 2 events in the first hour or at least 2 events in the second hour? e) Find the probability ( ) p t of exactly one event in a period of t hours. f) Find lim ( ) / t p t t ! and provide an interpretation of this limit. 4. In question 3, a) If W is the time until the first event, find ( 1) P W ! . b) If V is the time until the second event, find ( 2) P V ! ....
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