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.

1. (a)Find the expected number of undamaged chairs that will be delivered.
2. (b)What is the probability to meet the terms of the contract if 1050 chairs are shipped?
3. (c)What is the minimum number of chairs that should be shipped to meet the terms of
4. 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$.

1. (a)Find the expected price in 250 days (on the day number 250 starting from today).
2. (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
3. result of problem 4.