review_discrete

# However such a random variable can only be observed

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Unformatted text preview: n a tree leaf by an insect of a certain type is a Poisson random variable with parameter λ. However, such a random variable can only be observed if it is positive, since if it is 0 then we cannot know that such an insect was on the leaf. If we let Y denote the observed number of eggs, then P (Y = x) = P (X = x|X &gt; 0) where X is Poisson with parameter λ. Find E (Y ). 12 Discrete random variables - some examples Example 1 In a gambling game a person who draws a jack or a queen is paid \$15 and \$5 for drawing a king or an ace from an ordinary deck of ﬁfty-two playing cards. A person who draws any other card pays \$4. If a person plays the game, what is the expected gain? Example 2 A purchaser of electrical components buys them in lots of size 10. It is his policy to inspect 3 components randomly from a lot and to accept the lot only if all 3 components are nondefective. If 30 percent of the lots have 4 defective components and 70 percent have only 1, what proportion of lots does the purchaser reject? Example 3 An urn contains N white and M black balls. Balls are randomly selected with replacement, one at a time, until a black one is obtained. a. What is the probability that exactly n draws are needed? a. What is the probability that at least k draws are needed? Example 4 An electronic fuse is produced by ﬁve production lines in a manufacturing operation. The fuses are costly, are quite reliable, and are shipped to suppliers in 100-unit lots. Because testing is destructive, most buyers of the fuses test only a small number of fuses before deciding to accept or reject lots of incoming fuses. All ﬁve production lines usually produce only 2% defective fuses. Unfortunately, production line 1 suﬀered mechanical diﬃculty and produced 5% defectives during the previous month. This situation became known to the manufacturer after the fuses had been shipped. A customer received a lot produced last month and tested three fuses. One failed. a. What is the probability that the lot produced on line 1? b. What is the probability that the lot came from one of the four other lines? Example 5 To determine whether or not they have a certain disease, 100 people are to have their blood tested. However, rather than testing each individual separately, it has been decided ﬁrst to group the people in groups of 10. The blood samples of the 10 people in each group will be pooled and analyzed together. If the test is negative, one test will suﬃce for the 10 people. If the test is positive each of the 10 people will also be individually tested. Suppose the probability that a person has the disease is 0.10 for all people independently from each other. Compute the expected number of tests necessary for each group. Example 6 A standard deck of 52 cards has been reduced to 51 because an unknown card was lost. As a reminder a standard deck of 52 cards consists of 13 clubs (♣), 13 spades (♠), 13 hearts (♥), and 13 diamonds (♦). a. A card is selected at random from the...
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## This note was uploaded on 10/04/2012 for the course STATISTICS 100a taught by Professor Cristou during the Spring '10 term at UCLA.

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