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Negative sign will come not positive.
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(1 point) BONUS. Given that 1 Show that Σα-.re: whe , where Sxx-〉 (xi-x) /Sxx
2. Suppose Xi ~ N(8,02) where θ > 0. (a) Show that s--(x, Σ¡! xi) is a sufficient statistic of θ where X is the sample mean. (b) Is S minimal sufficient? (c) Can you find a non-constant function g(.) such that g(S) is an ancillary statistic?
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3 (10) Let = Re', z = re (0<r< R) be two complex numbers. Show the following identities hold: R2 2 OO = Re = 1 +2 C-z ΣΑ. R2 - 2rR cos (-0)r2 coS n(-e) n=1
2. (20pts) Let Xi,..., X be a random sample from a population with pdf f(x)--(1 , where θ > 0 and x > 1. (a) Carry out the likelihood ratio tests of Ho : θ-a, versus Hi : θ a-show that the likelihod ratio statistic corresponding to this test, A, can be re-written as Λ = cYne-ouY, where Y Σ:.. In (X), and the constant c depends on n and θο but not on Y. (b) Make a sketch of...
1.(c)
2.(a),(b)
5. Let Xi,..., X, be iid N(e, 1). (a) Show that X is a complete sufficient statistic. (b) Show that the UMVUE of θ 2 is X2-1/n x"-'e-x/θ , x > 0.0 > 0 6. Let Xi, ,Xn be i.i.d. gamma(α,6) where α > l is known. ( f(x) Γ(α)θα (a) Show that Σ X, is complete and sufficient for θ (b) Find ElI/X] (c) Find the UMVUE of 1/0 -e λ , X > 0 2) (x...
2. Show that W can be written as where U is the number of pairs (Xi, Yj) with X, < Y,. In other words n m U=ΣΣ1," where ,j -(0 otherwise. i=1 j=1 Hint: Let Yi),Ya),... , Ym) be the order statistics for the y-sample. Then U is the number of pairs (Xi,Yu)) with Xi 〈 YG). For fixed j , the number of Xi with Xi 〈 Yu) is just the rank of Y (j) minus the number of...
a power * A random process X(t) has Spectral density given by Sxx (w) = Su-we luk 6 Otherwise . Determine the average power.
(10 pts). Show that the within-cluster variation (WCV) satisfies k=1 C(i)=k,C(j)-k equals to Σ Σ 1x,-제12, where X,-1 Σα i) =k Xi ん
Let (xi , f(xi)), i = 0, . . . , 3, be data points, where xi = i
+ 2, for i = 0, . . . , 3. Given the divided differences f[x0] = 1,
f[x0, x1] = 2, f[x0, x1, x2] = −7, f[x0, x1, x2, x3] = 9, add the
data point (0, 3), find a Newton form for the Lagrange polynomial
interpolating all 5 data points.
3. (25 pts) Let (r,, f()), 0,3, be data...
Given three random variables Xi, X2, and X such that X[Xi X2 X 20 -1 3 1 0.5 1 E [X]-μ | 0 | and var(X)=Σー| 0 0.5 | com pute: 2 c) var(X2-X3 (d) var(X2 + X3) (e) cov(4X2 +X1,3Xi - 2X3)
(1 point BONUS) Show that the sum of the z-scores of a data set has to be zero. That is, show that