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- Let x1, x2, ... , X, be a random sample from N(u, o2), and let 0 = (6x1 - 2x2) -(4x3 -3x4) %3D be an estimator of the parameter µ, calculate the bias of this estimator.Find a number c that satisfies the conclusion of the Mean Value Theorem for f(x) = x +x – 1 on the interval [0, 2] %3D V3 1 2. V3 V3 2. O V312. Please do all the development
- 7. a) Suppose that X is a uniform continuous random variable where 06. Suppose the moment generating function of X is M(t) = ² et + e²t+e³t. a) Find the mean of X. b) Find the variance of X. c) Find the pmf of X.Do #26Let Y₁, T₂, T3, T4, Is and X₁, Xe, Xq be Independent and normally From distributed random Sampits populations with mtans U₁ = 2 and 4₂ = 8 and Variances 0² = 5 and 0₂2₂²² = k respectively. Suppost P(x = √ > 10) = 0.02275, find the of 2 0₂²² = k· that Valueb) Let X₁, X₂,..., X and Y₁, Y₂, ..., Ym be random samples from populations with moment generating 25 functions Mx₁(t) = ³t+t² and My(t) = (₁-¹)²5, respectively. i) Find the sampling distribution of the statistic W = X₁ + 2X₂ − X3 + X4 + X5. ii) What is the value of the sample size n, if P[Σ1(X¡ − X)² > 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|Ỹ - µy| ≥10) < 0.04?Let X be a random variable. Suppose we try to use a constant real number â to guess the value of X. Let E(X – î)² be the mean squared error (MSE) of this estimation. (i) By writing X − â = (X − E X) + (EX − 2), show that E(X)² = Var(X) + (EX - 2)². = (ii) Conclude that the MSE is minimized for î Var(X). EX and the minimum MSE equals to20. Let X1, X2, ables, and set X, be independent, Exp(a)-distributed random vari- Y1 = X(1) and Y = X(k) – X(k-1); for 2Let X, Y, Z be the three assets, with VAR(X) = VAR(Y) = VAR(Z) = 1. Z is independent from X and Y , and COV(X, Y) = 0.2 .Assuming no short selling, what is the portfolio weight on Z that gives the smallest variance? (Hint: since Z is independent with X and Y, you may express wx and wy as a function of 1- wz , and then find wz.)Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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