Q 8.4. Let X₁, X2, Y₁ and Y2 be independent random variables each having a Gaussian dis-tribution. Suppose that E[X₁] = E[X₂]µ₁, that E[Y₁] = E[Y₂]μ2 and that var(X₁)var (X₂) = var (Y₁) = var(Y₂) = o² > 0.Let X = (X₁ + X₂) and Ỹ = ½ (Y₁ + Y₂). Define a random variable S² viaS²=2Σ(Xi − X)² + Σ(Y₂ − Ỹ)²ΣΥi-n²)i=11 (2₁xi=1=(1/√2 1/√20The aim of this question is to describe the joint distribution of X - Y and S²(a) Let U be a 4 4 matrix whose first two rows are001/√2 1/√₂)-Choose two further rows so that U is an orthogonal matrix. [Hint: it's a good plan to useplenty of zeros!](b) Let W be the 4-dimensional random vector W = (X₁, X2, Y₁, Y2), and define V to be therandom vector V = UW. Find the mean vector and variance-covariance matrix of V.(c) Show that you can write S² as a function of V3 and V4 and that X - Y is a function of V₁and V₂. Use this to describe the joint distribution of X - Y and S².

a) To construct an orthogonal matrix U from the given rows, we need to choose the other two rows…

Answered: Q 8.4. Let X₁, X2, Y₁ and Y2 be…

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Topics In Analytic Geometry

Section6.4: Hyperbolas

Problem 5ECP: Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.

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Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.
Q 8.4. Let X₁, X2, Y₁ and Y₂ be independent random variables each having a Gaussian dis- tribution. Suppose that E[X₁] = E[X₂] = µ₁, that E[Y₁] = E[Y₂] = µ2 and that var(X₁) var (X₂) = var(Y₁) = var(Y₂) = o² > 0. Let X = (X₁ + X₂) and Ỹ = ½ (Y₁ + Y₂). Define a random variable S² via 8² - 1 (21X₁-RP+ (M-91²) 82 = 2 2 i=1 i=1 The aim of this question is to describe the joint distribution of X – Ỹ and S² (a) Let U be a 4 x 4 matrix whose first two rows are (1/1/2 √√2 1/√2 0 0 1/√ 2 1/√2) = Choose two further rows so that U is an orthogonal matrix. [ Hint: it's a good plan to use plenty of zeros!] (b) Let W be the 4-dimensional random vector W = (X₁, X2, Y₁, Y₂), and define V to be the random vector V = UW. Find the mean vector and variance-covariance matrix of V. (c) Show that you can write S² as a function of V3 and V4 and that Ă – Ỹ is a function of V₁ and V₂. Use this to describe the joint distribution of X – Y and S².
B) Let X1,X2, .,Xn be a random sample from a N(u, o2) population with both parameters unknown. Consider the two estimators S2 and ô? for o? where S2 is the sample variance, i.e. s2 =E,(X, – X)² and ở² = 'E".,(X1 – X)². [X = =E-, X, is the sample mean]. %3D n-1 Li%3D1 [Hint: a2 (п-1)52 -~x~-1 which has mean (n-1) and variance 2(n-1)] i) Show that S2 is unbiased for o2. Find variance of S2. ii) Find the bias of 62 and the variance of ô2. iii) Show that Mean Square Error (MSE) of ô2 is smaller than MSE of S?. iv) Show that both S2 and ô? are consistent estimators for o?.
If X₁, X2,..., Xn constitute a random sample of size n from an exponential population, show that X is a sufficient estimator of the parameter 0.
Suppose X and Y are two random variables with E[X] = 1, Var (X) = 4, E[Y ] = -1, Var (Y) 4, and Cov (X, Y) = 1. Find the standard deviation of (X - Y). = (a) 2 (b) √2 (c) 6 (d) √6 (e) None of the above
4. Suppose that X has pdf f(x) = 3x² for 0 < x< 1. Find the pdf of the random variable Y = VX.
Consider two independent exponential random variables X1 and X2 with parameter lambda=1. LetY1 = X1 Y2 = X1 + X2. Find the MMSE estimate of Y1 using Y2.
4. Assume that X is an exponential random variable. Suppose further that Var(X) = 5. E(X). (a) Find the parameter 0> 0. (b) Compute Var(X + 4). (c) Compute P(X > 15[X > 11).
4.Suppose that X is a random variable for which E(X)=µ and Var(X)=o². Show that E(X(X-1)]=µ(µ-1)+o?
2.1.16. Let X1, X2 be two random variables with joint pdf f (x1, x2) = x1 exp{-x2}, for 0 < x1 < x2 < ∞, zero elsewhere. Determine the joint mgf of X1, X2. Does M(tı, t2) = M(t,0)M(0, t2)? %3D
2.5.8) Suppose that Y is a continuous random variable whose pdf is given by f(y) = {K (4y = 2y²), 0 1). (c) Find F(y).
2. Suppose n>1. Let X₁, X₂, Xn be independently and identically distributed (iid) with mean and variance o². Let Xn = Ei=1 X₁. Find (a) Var(4X₁-3X₂) (b) E[X2] (c) Var (X₁ + X). (Beware: Are X, and X, independent? Hint: Write X, as a linear combination of X, and Xn-1-)

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