Math135_hw4_soln

# Math135_hw4_soln - Math 135, Fall 2011: HW 4 This homework...

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Math 135, Fall 2011: HW 4 This homework is due at the beginning of class on Wednesday October 5th, 2011 . You are free to talk with each other and get help. However, you should write up your own solutions and understand everything that you write. Problem 1 (p.121 #3) . The chance of getting 25 or more sixes in 100 rolls of a die is 0 . 022. If you rolled 100 dice once every day for a year, ﬁnd the chance that you would see 25 or more sixes: a) at least once; b) at least twice. SOLUTION. Each trial consists of rolling 100 dice. We deem the event { 25 or more sixes in rolls of 100 dice } a success, and we are told its probability is p = 0 . 022. Since we are repeating the trial n = 365 times, we can use the Poisson approximation with μ = np , which in this case is μ = (365)(0 . 022). We get P (at least one success) 1 - e - 8 . 03 P (at least two successes) 1 - ± e - 8 . 03 + e - 8 . 03 8 . 03 1 1! ² Problem 2 (p.122 #7) . Let S be the number of successes in 25 independent trials with probability 1/10 of success on each trial. Let m be the most likely value of S . a) Find m . SOLUTION. The mode of a binomial B ( n,p ) distribution is m = b np + p c , so for np + p = 2 . 6, we get m = 2. b) Find P ( S = m ) correct to three decimal places. SOLUTION. P ( S = m ) = ³ 25 2 ´ (0 . 1) 2 ( . 9) 23 0 . 266 c) Compute the normal approximation to P ( S = m ). SOLUTION. Here, we want to calculate P ( S = m ) = P (2 - 0 . 5 S 2 + 0 . 5) = P 1 . 5 - 2 . 5 p (25)(0 . 1)(0 . 9) S - 2 . 5 p (25)(0 . 1)(0 . 9) 2 . 5 - 2 . 5 p (25)(0 . 1)(0 . 9) ! Φ(0) - Φ - 1 p (25)(0 . 1)(0 . 9) ! 0 . 5 - 0 . 2546 = 0 . 2454 1

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d) What is the value of the Poisson approximation to P ( S = m )? SOLUTION. Here we use μ = np = 2 . 5, and get P ( S = 2) = e - 2 . 5 (2 . 5) 2 2! 0 . 2565 e) Repeat parts (a), (b), (c) and (d) for n = 2500 instead of 25, and determine which approxi- mation, the normal or the Poisson, is better. Here p is still not all that small, and n is very large. The Binomial is about 0 . 0266. The normal ( 0 . 02658) and Poisson ( 0 . 0252) approximation are fairly close to one another; the normal approximation is slightly better. Depending on how people rounded their answers, it’s possible to conclude that the Poisson approximation is better. If your reasoning is clear, you will get credit either way. f) Repeat part (e) for n = 2500 and p = 1 / 1000 instead of p = 1 / 10. SOLUTION. Since
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## Math135_hw4_soln - Math 135, Fall 2011: HW 4 This homework...

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