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Normal approximation to binomial distribution
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4.86. Find the probability that 200 tosses of a coin will result in (a) between 80 and 120 heads inclusive, (b) less than 90 heads, (c) less than 85 or more than 115 heads, (d) exactly 100 heads. 4.87. Find the probability that a student can guess correctly the answers to (a) 12 or more out of 20, (b) 24 or more out of 40, questions on a true-false examination. 4.88. A machine produces bolts which are 10% defective. Find the probability that in a random sample of 400 bolts produced by this machine, (a) at most 30, (b) between 30 and 50, (c) between 35 and 45, (d) 65 or more, of the bolts will be defective. 4.89. Find the probability of getting more than 25 sevens in 100 tosses of a pair of fair dice.
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The Poisson distribution
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4.90. If 3% of the electric bulbs manufactured by a company are defective, find the probability that in a sample of 100 bulbs, (a) 0, (b) 1, (c) 2, (d) 3, (e) 4, (f) 5 bulbs will be defective. 4.91. In Problem 4.90, find the probability that (a) more than 5, (b) between 1 and 3, (c) less than or equal to 2, bulbs will be defective. 4.92. A bag contains 1 red and 7 white marbles. A marble is drawn from the bag, and its color is observed. Then the marble is put back into the bag and the contents are thoroughly mixed. Using (a) the binomial distribution, (b) the Poisson approximation to the binomial distribution, find the probability that in 8 such drawings, a red ball is selected exactly 3 times. 4.93. According to the National Office of Vital Statistics of the U.S. Department of Health and Human Services, the average number of accidental drownings per year in the United States is 3.0 per 100,000 population. Find the probability that in a city of population 200,000 there will be (a) 0, (b) 2, (c) 6, (d) 8, (e) between 4 and 8, (f) fewer than 3, accidental drownings per year. 4.94. Prove that if X1 and X2 are independent Poisson variables with respective parameters 1 and 2, then X1 X2 has a Poisson distribution with parameter 1 2. (Hint: Use the moment generating function.) Generalize the result to n variables.
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Multinomial distribution
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4.95. A fair die is tossed 6 times. Find the probability that (a) 1 one , 2 twos and 3 threes will turn up, (b) each side will turn up once.
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CHAPTER 4 Special Probability Distributions
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4.96. A box contains a very large number of red, white, blue, and yellow marbles in the ratio 4:3:2:1. Find the probability that in 10 drawings (a) 4 red, 3 white, 2 blue, and 1 yellow marble will be drawn, (b) 8 red and 2 yellow marbles will be drawn. 4.97. Find the probability of not getting a 1, 2, or 3 in 4 tosses of a fair die.
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The hypergeometric distribution
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4.98. A box contains 5 red and 10 white marbles. If 8 marbles are to be chosen at random (without replacement), determine the probability that (a) 4 will be red, (b) all will be white, (c) at least one will be red. 4.99. If 13 cards are to be chosen at random (without replacement) from an ordinary deck of 52 cards, find the probability that (a) 6 will be picture cards, (b) none will be picture cards. 4.100. Out of 60 applicants to a university, 40 are from the East. If 20 applicants are to be selected at random, find the probability that (a) 10, (b) not more than 2, will be from the East.
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The uniform distribution
4.101. Let X be uniformly distributed in 2 x 2 Find (a) P(X 1), (b) P( Z X 1Z
1 2 ).
4.102. Find (a) the third, (b) the fourth moment about the mean of a uniform distribution. 4.103. Determine the coefficient of (a) skewness, (b) kurtosis of a uniform distribution. 4.104. If X and Y are independent and both uniformly distributed in the interval from 0 to 1, find P( Z X YZ
1 2 ).