Question B : Consider the linear regression model questions

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Law of Large Numbers and Central Limit Theorem 1. Let X1, ..., X, be i.i.d. with common pdf f(x) = 3e-3(#-9), for x > 0. Let Y, = min( X1, . . ., Xn). (a) Noticing that P(Y, > y) = P(X1 > I, . .., Xn > I) = P(X1 > x)", find the pdf of Yn. (b) Notice that Y > 0, so the event (|Y, - 0| > (} = (Yn - 0 > E}, Show that Y, 0 in probability.

conceptual basis for our investigation". Assume that annual peak ﬂows in the river are statistically independent" from year to year. There are 160 years of daily measur-Ilts from which we can compute 160 annual peak ﬂows". a. Given lﬁU—years of annual peak flow data: {XITX2T.",X15,J}, what is the probability that at least one of these observations meets or excel-ids the IUD—year ﬂow"? b. Given lﬁﬂ—years of annual peak flow data: {X1,Xg,u.,xim}, what is the probability that the maximum of these 160 observations meets or exceeds the 100~year ﬂow? 1:. Given lﬁﬂ-years of annual peak ﬂow data: {leX;,.u,X1m}, what is the probability that the maximum of these 160 observations does NOT meet or exceed the lDﬂ-year ﬂow? d. On the average, how many years of annual peak flow data need to be collected before one of the observations meets or exceeds the I'm-year ﬂow? e. How many years of annual peak flow data need to be collected to be 95 percent certain that at least one of the observations elects or exceeds the IUD-year How? f. Let N represent the number of years of annual peak ﬂow data that you have available. Let I: represent the probability that the maximum of these N obsen'ations meets or exceeds the IOU-year ﬂow' Plot 13 {on the ordinate - vertical axis) versus H (on the abscissa - horizontal axis) for N = 1+ 2. . . . .1000. Use a leg-scale for both axes".

QUESTION 3 Consider the simple linear regression model, 1!: BO + 311+ u. Which of the following statements is correct? SLRJ , the ﬁrst assumption of the simple Linear regression model, is about bow the data used to estimate the parameters fig and [31 O a-are obtained from a random sample. SLFl.2. the second assumption of the simple linear regression model, is about the sample outcomes on x, {x}, i = 1, ..., n} not all ' being the same value. Ob SLFl.3. the third assumption of the simple linear regression modeL is about the population model being linear in the parameters 0 a. B0 and B1. 0 d. SLFH. the fourth assumption of the simple linear regression model. is about the zero conditional mean ofthe error term at.

4. Consider a discrete-time Markov chain with the following probability transition matrix 0 10 0 0 7 1-3-y P = 0 1 0 Is it possible to choose values for a and y so that the Markov chain has the following properties? In each case, state the values of a and y, or give a brief reason why it is not possible. (a) The Markov chain has period 2. [2) (b) The Markov chain is reducible. (c) The Markov chain has at least one transient state. GNN (d) The Markov chain has invariant distribution (1/4, 1/4, 1/4, 1/4).

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