The income of individuals in a given population is independently and identically distributed with a N (µ, 100) distribution. A researcher sampled the income of 25 individuals from such populations, and calculated the average Y = (Ex) /25. Which of the following is correct? O None of the above O Pr(Y <) = .5 O Pr( S) = .5 O Pr(Y Sp) = .9 O Pr(Y Sp) = .1

The Index of Industrial Production (IPt) is a monthly time series that measures the quantity of industrialcommodities produced in a given month. This problem uses data on this index for the United States. Allregressions are estimated over the sample period 1986:M1–2013:M12 (that is, January 1986 through December2013). Let Yt = 1200 × ln(IPt/IPt−1).(a) A forecaster states that Yt shows the monthly percentage change in IP, measured in percentage points perannum. Is this correct? Why?(b) Suppose she estimates the following AR(4) model for Yt:Yˆt = 0.787 + 0.052Yt−1 + 0.185Yt−2 + 0.234Yt−3 + 0.164Yt−4Use this AR(4) to forecast the value of Yt in January 2014, using the following values of IP for July 2013 throughDecember 2013:Table 1:Date 2013 M:7 2013 M:8 2013 M:9 2013 M:10 2013 M:11 2013 M:12IP 99.016 99.561 100.196 100.374 101.034 101.359(c) Worried about potential seasonal fluctuations in production, she adds Yt−12 to the autoregression. Theestimated coefficient on Yt−12 is 0.061, with…

iid *1. Let Y1,..., Yn * uniform(0, 1). Find the distribution of their geometric mean: U = i=1 (Hint: consider first finding the distribution of –n log U.) - N (µ, o2). Find the distribution of Y and show that P(Y s e") = 0.5 (i.e., e" is *2. Let log Y the median of Y).

If W denotes the weight (in pounds) of an individual and t denotes time (in weeks), then dW/dt is the rate of weight gain or loss (in lb/wk). Suppose that a person weighed 220 pounds 10 weeks ago, and now weighs 198 pounds. Using the Mean Value Theorem, show that their rate of rate loss must have exceeded 2 lb/wk at some point during this 10 week period.