The following is a sufficient condition for the central limit theorem: If the random variables X1, X2, . . . , Xn are independent and uniformly bounded (that is, there exists a positive constant k such that the probability is zero that any one of the random variables Xi will take on a value greater than k or less than – k), then if the variance of Yn = X1 + X2 + · · · + Xn becomes infinite when n. ∞, the distribution of the standardized mean of the Xi approaches the standard normal distribution. Show that this sufficient condition holds for a sequence of independent random variables Xi having the respective probability distributions
Answer to relevant QuestionsA random sample of size 64 is taken from a normal population with µ = 51.4 and σ = 6.8. What is the probability that the mean of the sample will (a) Exceed 52.9; (b) Fall between 50.5 and 52.3; (c) Be less than 50.6? The actual proportion of men who favor a certain tax proposal is 0.40 and the corresponding proportion for women is 0.25; n1 = 500 men and n2 = 400 women are interviewed at random, and their individual responses are looked ...If S1 and S2 are the standard deviations of independent random samples of sizes n1 = 61 and n2 = 31 from normal populations with s21 = 12 and s22 = 18, find P(S21 / S22 > 1.16). Use the result of Exercise 8.56 to find the probability that the range of a random sample of size n = 5 from the given uniform population will be at least 0.75. Basing their decisions on pessimism as in Example 9.2, where should (a) Ms. Cooper of Exercise 9.12 make her reservation; (b) The truck driver of Exercise 9.13 go first? Example 9.2 With reference to Example 9.1, suppose ...
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