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Let Y be the number of successes throughout n independent repetitions of a random experiment with probability of success . Determine the smallest value of n so that 

Over the years, the percentage of candidates passing an entrance exam to a prestigious law school is 20%. At one of the testing centers, a group of 50 candidates take the exam and 20 pass. Is this odd? Answer on the basis that X ≥ 20 where X is the number that pass in a

Let  be two independent random variables having gamma distributions with parameters 

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