Example 5: Let X1, X2 , X3 and X, be independent random variables such thatE (X;) = µ and Var (X;) = o? for i= 1, 2, 3, 4.X1+ X2+ 2X3+ X4Given that (i) Y=Examine whether Y is unbiased estimator of u4a X1 +X2 + X3(ii) If T:find 'a' such that T is an unbiased estimator.%3D(iii) Which is the best estimator?

Answered: Image /qna-images/answer/6af0466d-8346-42e1-a22a-84f8706c0d11.jpg

Answered: Example 5: Let X1, X2 , X3 and X4 be…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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4. Show that if X and Y are independent random variables, then E(g(X)| X = ro) = 9(ro).
If X and Y are independent random variables with E(X) = 10, Var(X) = 4, E(Y) = 12 and VAR(Y) = 3, If W =-2X +Y Find E(W) and STD (W) (a) (b) If X and Y are not independent, with E(X) =10, VAR(X) = 5, E(I) =18, VAR(Y) = 3 and CovX,Y) = -2. If W = X – 2Y Find E(W) and STD(W)
Suppose that Y₁, Y₂, ..., Ym is a random sample of size m from Gamma (a = 3, B = 0), where 0 is not known. Check whether or not the maximum likelihood estimator Ô is a minimum variance unbiased estimator of the parameter 8.
Example 10-3. Show that if X', Y’ are the deviations of the random variables X andY from their respective means, then 1 (1) r =1- 2N X'; Y;\ Ox Oy,
show that if X11, X12,..., X1, X21, X22,..., X2n₂ are independent random variables, with the first n constitut- ing a random sample from an infinite population with the mean μ₁ and the variance of and the other n2 constitut- ing a random sample from an infinite population with the mean μ2 and the variance o2, then 1 M2 (a) E(X₁-X₂) = μ₁ −μ2; 07 0 22 (b) var(X₁-X₂) = + n₁ n₂
Let X1, X2,..., X3 denote a random sample from a population having mean u and variance o?... Which of the estimators have a variance of 7 X1+X2++X, 7 2X1-X6+X4 2 3X1-X3+X4 2 2(X1+X2+.+X¬) 4 7
B5. Let X₁, X₂, ..., Xn be IID random variable with common expectation µ and common variance o², and let X = (X₁ + + X₂)/n be the mean of these random variables. We will be considering the random variable S² given by (a) By writing or otherwise, show that S² (b) Hence or otherwise, show that n S² = (x₁ - x)². = Ĺ(X₂ i=1 X₁ X = (X₁-μ) - (x-μ) = Σ(X; -μ)² - n(X - μ)². i=1 ES² = (n-1)0². You may use facts about X from the notes provided you state them clearly. (You may find it helpful to recognise some expectations as definitional formulas for variances, where appropriate.) (c) At the beginning of this module, we defined the sample variance of the values x₁, x2,...,xn to be S = 1 n-1 n i=1 ((x₁ - x)². Explain one reason why we might consider it appropriate to use 1/(n-1) as the factor at the beginning of this expression, rather than simply 1/n. B6. (New) Roughly how many times should I toss a coin for there to be a 95% chance that between 49% and 510/ of my nain toon land Honda?
2
Let X₁, X2, X3, Xn be a random sample with unknown mean EX; = µ, and unknown variance Var(X₂) = o². Suppose that we would like to estimate 0 = μ². We define the estimator as 2 • - (™)² - [ 2x]* Xk to estimate 0. Is an unbiased estimator of ? Why?
Three statistically independent random variables X₁, X₂ and X₂ have mean values X₁ =3, X₂ = 6 and X₂ = -2. Find the mean values of the following functions: 1 2 (a) (b) g(X₁, X₂, X3)= X₁ +3X₂ + 4X3 29 g(X₁, X₂, X3) = X₁X₂X₂ 29
Given that X and Y are independent random variables and E(X) = 4, E(Y) = 7, Var(X) = 3, Var(Y) = 6; find Var(8X - 2Y)
Example 3.17 Let X be a discrete random variable with range Rx = {0, 7, 5, ,"}, such that Px(0) = Px(;) = Px(5) = Px() = Px(x) = . Find Esin(X).

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