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- The analysis of tooth shrinkage by C. Loring Brace and colleagues at the University of Michigan’s Museum of Anthropology indicates that human tooth size is con-tinuing to decrease and that the evolutionary process has not yet come to a halt. In northern Europeans, for example, tooth size reduction now has a rate of 1% per 1000 years. a. If t represents time in years and y represents tooth size, use the condition that y = 0.99y0 when t = 1000 to find the value of k in the equation y = y0 ekt. Then use this value of k to answer the following questions. b. In about how many years will human teeth be 90% of their present size? c. What will be our descendants’ tooth size 20,000 years from now (as a percentage of our present tooth size)?For a certain insured, the distribution of aggregate claims is binomial with parameters m = 12 and q = 0.25. The insurer will pay a dividend, D, equal to the excess of 80% of the premium over claims (excess = 0.8 x premium - claims), if positive. The premium is 5. Calculate E[D].The best fit line for a time series, {yt : t = 1,2,...,518} is found to be f (t) = 3.4025t + 199.3715. Based on the best fit line, what is the forecast value, when t = 518?
- The analysis of tooth shrinkage byC. Loring Brace and colleagues at the University of Michigan’s Museum of Anthropology indicates that human tooth size is continuing to decrease and that the evolutionary process has not yetcome to a halt. In northern Europeans, for example, tooth sizereduction now has a rate of 1% per 1000 years.a. If t represents time in years and y represents tooth size, usethe condition that y = 0.99y0 when t = 1000 to find thevalue of k in the equation y = y0ekt. Then use this value of kto answer the following questions.b. In about how many years will human teeth be 90% of theirpresent size?c. What will be our descendants’ tooth size 20,000 years fromnow (as a percentage of our present tooth size)?The time for the first widget to be manufactured each morning is random between 1 and 3 seconds with a pdf of f(x)=1/162(x-3)^2(x+6) for 0 < x < 6What is the cumulative distribution function, F(x)? What is the probability of a widget being made earlier than 4 s?Answer to four decimal places What is the probability of a widget being made later than 1.3 s?Answer to four decimal placesFind the equation of the normal line to the graph of the function Y = Ln (X3 + 4), at the point (-1, 1.09)
- What is the Marginal PDF of f(x)=2? Is the event independent?A college professor never finishes his lecture before the end of the hour and always finishes his lectures within 2 min after the hour. Let X = the time that elapses between the end of the hour and the end of the lecture and suppose the pdf of X is as follows. The pdf is f(x) = kx^2 if 0 <= x <= 2, and 0 otherwise. (a) Find the value of k. (Enter your answer to three decimal places.) (b) What is the probability that the lecture ends within 1 min of the end of the hour? (Enter your answer to three decimal places.) (c) What is the probability that the lecture continues beyond the hour for between 30 and 90 sec? (Round your answer to four decimal places.)Let X denote the vibratory stress (psi) on a wind turbine blade at a particular wind speed in a wind tunnel. Suppose that X has the following Rayleigh pdf.f (x) = { (x/θ2) e−x2/(2θ2) x > 0 0 otherwise (a) If θ = 104, find the probability that the vibratory stress is between 102 and 378. (b) If θ = 104, then 71% of the time the vibratory stress is greater than what value?
- A train’s arrival time at the station is T minutes past noon, where the PDF of T is given below. Given that the train hasn’t arrived by 12:05, what is the probability that it arrives by 12:10?T has an exponential distribution such that P(T≤2)=2P(T>4) What is Var(T)=?Suppose X1, . . . , Xn ∼ Exponential(λ) is a set of n observations drawn independently from an Exponential distribution. (e) Take the second partial derivative of the score function. (f) Check to make sure this value is negative to ensure that the log-likelihood function is concave down.