Suppose X following a binomial distribution with n and p. Then, X has the following pmf: Px(x)=(-)p(1 - p)" for x = 0, 1, … … …, n, (1) with the mean E[X] = np and variance Var[X] = np(1 - p). It is known that the normal approximation to the binomial is possible when the number of trials n is large enough, i.e., np ≥ 10 & n(1 - p) ≥ 10. That is, large n px(x)N(np, np(1 − p)). (2) Now, suppose a patient guy rolled a fair die 300 times and classified 'Success' as rolling a 6, and 'Failure' as rolling any other number in each trial. Please answer the following questions based on the models in (1) and (2). (i) We first model this experiment with a binomial distribution in (1). What are the parameter n and p? Save the values in the variables n and p. (ii) Compute the following probabilities using pbinom(): ⚫the number of having "six" is at least 44. ⚫ the number of having "six" is at most 50. the number of having "six" is at least 30 and at most 65. (iii) Based on the result in (2), compute the three probabilities in #1-(ii) using pnorm()¹, and comment on the quality approximations. Are those acceptable? Continuity correction is not required for this problem.

Suppose X following a binomial distribution with n and p. Then, X has the following pmf: Px(x)=(-)p(1 - p)" for x = 0, 1, … … …, n, (1) with the mean E[X] = np and variance Var[X] = np(1 - p). It is known that the normal approximation to the binomial is possible when the number of trials n is large enough, i.e., np ≥ 10 & n(1 - p) ≥ 10. That is, large n px(x)N(np, np(1 − p)). (2) Now, suppose a patient guy rolled a fair die 300 times and classified 'Success' as rolling a 6, and 'Failure' as rolling any other number in each trial. Please answer the following questions based on the models in (1) and (2). (i) We first model this experiment with a binomial distribution in (1). What are the parameter n and p? Save the values in the variables n and p. (ii) Compute the following probabilities using pbinom(): ⚫the number of having "six" is at least 44. ⚫ the number of having "six" is at most 50. the number of having "six" is at least 30 and at most 65. (iii) Based on the result in (2), compute the three probabilities in #1-(ii) using pnorm()¹, and comment on the quality approximations. Are those acceptable? Continuity correction is not required for this problem.

Operations Research : Applications and Algorithms

4th Edition

ISBN:9780534380588

Author:Wayne L. Winston

Publisher:Wayne L. Winston

Chapter12: Review Of Calculus And Probability

Section12.7: Z-transforms

Problem 3P

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