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- Suppose that a firm's production function is given by: F(K, L) = 4KL – 2 min{K², L²}, where K represents capital input and L labor input. As a consequence, F(K, L) = 4KL-2L² if L K. (c) MP₁ = 4L if L K. (d) MP₁ = 4K – 4L if L K. = = 4K 4L if L > K.Suppose in a practical situation, γ = min(X) >0 and X - γ has a Weibull distribution. LetX = the time (in 10-1 weeks) from shipment of a defective product until the customerreturns the product. Suppose that the minimum return time is γ = 3.5 and that the excessX – 3.5 over the minimum has a Weibull distribution with parameters α = 2 and β = 1.5.a. What is the cumulative distribution function (cdf ) of X ?b. What are the expected return time and variance of return time?Qs. A random sample X1, X2, ..., X, is drawn from a Pareto population with pdf f(x; 0, v) = 0+1;* > v,0 > 0, v > 0. (a) Find the MLE of 0 and v. (b) Perform a LRT of Ho : 0 = 1, v is unknown, versus H1 : 0 # 1, v is unknown. Show that the critical region of the test is of the form {T(x) c2}, where 0 < c1 < c2 and IT X; (min X;)" T = log (c) Show that, under Ho, 2T has a chi-squared distribution, and find its degrees of freedom.
- Find the generating function of the negative binomial mass function = (x =1) f(k) = p' (1-p)k-r, k = r,r +1,..., where 0 < p < 1 and r is a positive integer. Deduce the mean and variance.Consider the following initial value problem, in which an input of large amplitude and short duration has been idealized as a delta function.Suppose a high-tech shop opens at 9 am. During the release of a long-waited new generation smart phone people start arriving at the shop at 8 am and queue in front of the shop. The arrivals follow a non-homogeneous Poisson process with rate function λ(t) = 25t2 (t = 0 refers to 8 am and t = 1 to 9 am). (a) Find the distribution of the number of people waiting in front of the shop when the shop opens. (b) If we know that exactly 30 people are waiting in front of the shop at 9 am, what is the probability that exactly 20 of them arrived between 8.30am and 9 am? (c) Calculate the average waiting time of a person who is present at 9 am.
- 9. Suppose (X, Y) have the joint density f(x, y) = cxy, x, y ≥ 0; x+y≤ 1. • (a) Find the normalizing constant c. • (b) Are X, Y independent? • (c) Find the marginal densities and expectations of X, Y. • (d) Find the conditional expectation of Y given X = x.2. We would like to fit a linear regression estimate to the dataset {(x®,y@),(x), y),., (x(N), g/N)} with x e RM by minimizing the ordinary least square (OLS) objective function: N M -Συ, .(i) J(w): j=1 Specifically, we solve for each coefficient wk (1< k < M) by deriving an expression of Wk from the critical point J(w) the dataset (x(1), y(1)), (x(2), y(2)), . 0. What is the expression for each wk in terms of … , (x(^), y(N) and w1, , wk-1, Wk+1; *** , WM? .. .. Select one: E, (y() –D,-1,j+k W;x;") i=D1 Wk = =1 O WkConsider the Keynesian consumption function Yt = B₁ + B₂x2t + &t where yt is per capita consumption, and x2+ is per capita income. The coefficient ₂ is interpreted causally as the marginal propensity to consume, and we expect 0Suppose that a constant voltage source of 10 V supplies a current I (in mA) through a resistive load with a stochastic resistance R. (in k) defined by the PDF as follows. fR (r) B = = C = -{1 1500r750r²- 742.5, 0.9≤r≤1.1 elsewhere Knowing from ohm's law that I = Ꭱ the following PDF: fi(i) = { ¦ (Ai + Bi² + C), A = 0₂ and I the derived distribution of I is given by 100 111. (8 points) Suppose that Y1,..., Yn is a random sample from a population whose density is f (y\0) = 30°y¬4, y > 0, for some parameter 0 > 0. As we showed in class, the maximum likelihood estimator of 0 is ÔMLE min{Y1,..., Yn} and whose density function is .—Зп—1 fôMLE (7) = 3n0³nx provided x > 0. As shown on Midterm #1, Зп — 1 Зп — 1 OMLE E ) min{Y1, · · . , 3n 3n is an unbiased estimator of 0. Determine the density function of 0. Note that this describes the sampling distribution of Ô.8. (a) Assume process X, satisfies X,-aX-1+ Z where Z, is WN(0, a?) and Ja< 1. Use the Yule-Walker equation to find the best linear predictor P(X+iXn Xn-1). Find the mean-square error of the predictor. = Z, + aZ,-1 + bZ,-2, where Z, is WN(0, a). Use the (b) Assume process X, satisfies X, Durbin-Levinson algorithm to find the linear predictor P(X+i|X Xn-1)-Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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