Use the uniqueness of moment-generating functions to give the distribution and the mean of a random variable with moment-generating function m(t) = (0.9et + 0.1)6.
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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.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)?In an effort to make the distribution of income more nearly equal, the government of a country passes a tax law that changes the Lorenz curve from y = 0.98x2.1for one year to y = 0.32x2 + 0.68x for the next year. Find the Gini coefficient of income for both years. (Round your answers to three decimal places.) after before
- The lifetime of a certain type of TV remote control is given by Y . Suppose Y has approximately exponential distribution with mean 8 years. c) What should the warranty period for these remote controls be if the manufacturer wants 85% of the remote controls to last beyond the warranty period? d) What is the moment generating function of Y.If X is exponentially distributed with parameter λ and Y is uniformly distributed on the interval [a, b], what is the moment generating function of X + 2Y ?Determine the upper-tail critical value tα/2 in each of the following circumstances.
- 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 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?Suppose that the Lorenz Curve for country A is given by yA = x 2.2 and for country B is yB = 0.3x 2 + 0.7x. Use the Gini index to determine which of the two countries has greater inequality in its income distribution.