The multinomial distribution

Among typical july days in tampa 30 percent have

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Unformatted text preview: to n. Example 1 With the recent emphasis on solar energy, solar radiation has been carefully monitored at various sites in Florida. Among typical July days in Tampa, 30 percent have total radiation of at most 5 calories, 60 percent have total radiation of at most 6 calories, and 100 percent have total radiation of at most 8 calories. A solar collector for a hot water system is to be run for 6 days. Find the probability that 3 days will produce no more than 5 calories each, 1 day will produce between 5 and 6 calories, and 2 days will produce between 6 and 8 calories. What assumptions must be true for your answer to be correct? Show work. Let X – number of days having total radiation of at most 5 calories (5 or less), Y = number having total between 5 and 6 W =total between 6 and 8. Then (X,Y,W) have a multinomial distribution with parameters, n=6, px =0.3, py=0.6 0.3=0.3 And pw = 1 0.6 =0.4 Using the multinomial formula, we get P(X=3, Y=1, W=2)=[ 6!/(3! 1! 2!)] (0.3)3 (0.3)1 0.42 =0.07776 Example 2 A warehouse contains TV sets, of which 5% are defective, 60% are in working condition but used, and the rest are brand new. What is the probability that in a random sample of five TV sets from this warehouse, there are exactly one defective and exactly two brand new sets? Show work. P(X=1, Y=2, W=2)= 5! (0.05)(0.6) 2 (0.35) 2 = 0.066 1!2!2!...
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