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Let denote time until failure of a device and let denote the hazard function of .

Let X have an exponential distribution. (a) For  and , show that Hence, the exponential distribution has the memory less property. Recall from Exercise 3.1.9 that the discrete geometric distribution has a similar property. (b) Let  be the cdf of a continuous random variable Y . Assume that   Exercise 3.1.9 If  is the unique mode of a distribution that

Write an R function that returns the value for a specified x when is the Weibull pdf given in expression (3.3.12). Next write an R function that returns the associated hazard function Obtain side-by-side plots of the pdf and hazard function for the three cases: and and and and .

In Example 3.3.5, a page of plots of β pdfs was discussed. All of these pdfs are mound shaped. Obtain a page of plots for all combinations of α and β drawn from the set .25, .75, 1, 1.25. Comment on these shapes. Example 3.3.5

If

If  by using either Table II of Appendix D or the R command qnorm. Table II

If X is  find  and  pnorm. Table II

1. Evaluate  2. Determine the 90th percentile of the distribution, which is N(65, 25). 3. If  2. Let the random variable X have the pdf (a) Find the mean and the variance of X. (b) Find the cdf and hazard function of X.

If Show that the constant c can be selected so that , satisfies the conditions of a normal pdf. If . Show that the graph of a pdf has points of inflection at and .

Let If X is N(1, 4), compute the probability If X is N(75, 25), find the conditional probability that X is greater than 80 given that X is greater than 77. See Exercise 2.3.12. Exercise 2.3.12 Let and denote, respectively, the pdf and the cdf of the random variable X. The conditional pdf of ,