Question
Problem 1.Find the following probabilities using the normal approximation for the binomial distribution. (a)XBinom(100,0.5), P(X47) = ? (b)XBinom(1000,0.4), P(X390) = ? (c)XBinom(500,0.7), P(|X350| 10) =
Problem 1.Find the following probabilities using the normal approximation for the binomial distribution.
(a)XBinom(100,0.5), P(X47) = ?
(b)XBinom(1000,0.4), P(X390) = ?
(c)XBinom(500,0.7), P(|X350| 10) = ?
Problem 2.Company "O'Bender" has got a contract to deliver 1000 chairs. During transportation, a chair may be damaged with probability 5%. Assume chairs are damaged independently. To be sure to meet the terms of the contract, "O'Bender" plans to ship 1050 chairs.
- (a)Find the expected number of undamaged chairs that will be delivered.
- (b)What is the probability to meet the terms of the contract if 1050 chairs are shipped?
- (c)What is the minimum number of chairs that should be shipped to meet the terms of
- the contract with probability at least 90%?
Use the normal approximation of the binomial distribution to solve this problem.
Problem 3.The following quality control test is going to be implemented at a factory. A batch ofnproduced details will be selected and inspected. Let the number of defective details in this batch be denoted bym, and letp=m/nbe the proportion of defective details. The production line will be stopped for repair ifp >3%.
Suppose that the equipment breaks down and the rate of defective details becomes 4%. If a batch ofn= 1000 details is inspected, what is the probability that the test will be failed? Use the normal approximation.
Problem 4.Suppose random variablesX1, X2, . . . , Xnare independent, equal in dis- tribution, and each assumes only two values,aandb, with probabilitiespand 1prespectively. LetSn=X1+. . .+Xn. Using the normal approximation for the binomial distribution, derive the formula for P(Snx) whennis large.
Problem 5.Suppose the price of the stock of a some company is modeled by the fol- lowing simple model: each day with probability 1/2 it can either increase by 2% of the current price, or decrease by 2% of the current price; assume that daily price changes are independent. Suppose the price of 1 share of stock of the company today is 100$.
- (a)Find the expected price in 250 days (on the day number 250 starting from today).
- (b)Find the probability the the price on that day will be lower than the price today. Use the normal approximation.Hint:1.02 =eln1.02, 0.98 =eln0.98, and apply the
- result of problem 4.
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