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- Gertrude owns a movie theatre in the town of Northern Millstone. Assume the following to be true. If a resident of Northern Millstone patronizes Gertrude’s movie theatre on any given day, the probability that this resident will patronize Gertrude’s movie theatre the next day is .05. If a resident of Northern Millstone does not patronize Gertrude’smovie theatre on any given day, the probability that this resident will not patronize Gertrude’s movie theatre the next day is .9. Find the equilibrium vector for the Markov chain described in this problem. (An equilibrium vector is also known as a stationary matrix.) Express the elements of the equilibrium vector as fractions.If there are no vectors in the set S = {a1; a2; a3} from R^3 who is a multipleof one of the other vectors. Is the set S linearly independent or linearly dependent?We fit a ridge regression model to some data for λ=10 and λ=100 and we obtain the coefficients vector estimates b1 and b2 respectively. Then b1 is either more sparse than b2, or as sparse as b2. b2 is either more sparse than b1, or as sparse as b1. The L2 norm of b2 is less or equal to the L2 norm of b1. The L2 norm of b1 is less or equal to the L2 norm of b2.
- If two vectors ? = 4? + 6? − 8? and ? = 6? − 8? + 4? are added together describe the characteristics of the resultingvector.Let hat(\beta ) be the (k+1)\times 1 vector of OLS estimates.(i) Show that for any (k+1)\times 1 vector b, we can write the sum of squared residuals asSSR(b)=hat(u)^(')hat(u)+(hat(\beta )-b)^(')x^(')x(hat(\beta )-b).{():} Hint: Write (y-xb)^(')(y-xb)=[hat(u)+x(hat(\beta )-b)]^(')[hat(u)+x(hat(\beta )-b)] and use the fact that{:x^(')u=0.}6 Consider the setup of the Frisch-Waugh Theorem.(i) Using partitioned matrices, show that the first order conditions (x^(')x)hat(\beta )=x^(')y can be written asx_(1)^(')x_(1)hat(\beta )_(1)+x_(1)^(')x_(2)hat(\beta )_(2)=x_(1)^(')yx_(2)^(')x_(1)hat(\beta )_(1)+x_(2)^(')x_(2)hat(\beta )_(2)=x_(2)^(')y.(ii) Multiply the first set of equations by x_(2)^(')x_(1)(x_(1)^(')x_(1))^(-1) and subtract the result from the second setof equations to show that(x_(2)^(')M_(1)x_(2))hat(\beta )_(2)=x_(2)^(')M_(1)y,where M_(1)=I_(n)-x_(1)(x_(1)^(')x_(1))^(-1)x_(1)^('). Conclude thathat(\beta )_(2)=(x_(2)^(¨)^(')x_(2)^(¨))^(-1)x_(2)^(¨)^(')y.Find basis for the set of vectors in R2 on the line y= nx . where n=9.