3. Refer to problem 49 in Chapter 3 of Devore\'s textbook. Suppose that a shipme

Question

3. Refer to problem 49 in Chapter 3 of Devore's textbook. Suppose that a shipment of goblets arrives and the inspector is interested in knowing how many goblets will need to be inspected until the first faulty one is found. Denote this random variable as Y (a) What is the distribution for Y. Also state the value of the parameter. (b) Find the probability that the fifth goblet checked is the first defective. (c) Find the probability that the inspector will have to check 5 or more goblets in order to find the first defective. (d) What is the expected number of goblets the inspector will have to check in order to find the first defective? What is the variance in the number of goblets inspected until a defective is found

Explanation / Answer

a) here probabilty of finding defective goblets p=0.1

hence probabilty of passed goblets =1-0.1 =0.9

as this is geometric distribution, hence probabilty that we get first defective in Yth trail =P(in first y-1 trail there is no defective*at yth trail we get defective)

=(1-p)y-1p

b)from above probabilty that 5th piece checked is defective =(1-p)4*p =(0.94*0.1 =0.06561

c)probabilty he has to check 5 or more to get first defective =P(in first 5 there is no defective) =(0.9)5 =0.59049

d)from geometric distribution, expected number of goblets =1/p =1/0.1 =10

variance =(1-p)/p2 =0.9/(0.1)2 =0.9/0.01 =90

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