Answer the following questions: a. Let X1, X2. . . . . Xn be i..d. random...
Answer the following questions: a. Let X1, X2, . . . , Xn be i.i.d. random vectors (a random sample) from Np(μ1, Σ). Find the distribution of X ̄ . Note: X ̄ = 1/n Xi . b. Refer to question (a). Consider the following two random variables: Q1 = 1′X ̄/1'1 and Q2 = 1′Σ−1X ̄/1′Σ−11 ̄ . Find the mean and variance of Q1 and Q2 .
Part (b) details needed. thanks Answer the following questions: a. Let X1, X2. . . . . Xn be i..d. random vectors (a random sample) from Mpță Σ). Find the distribution of X. Note: X-ri Xi. b. Refer to question (a). Consider the following two random variables: Q1 and 1'1 Q2-1'Σǐ8. Find the mean and variance of (i and (2 1
3. Let X1, . . . , Xn be iid random variables with mean μ and variance σ2. Let X denote the sample mean and V-Σ,(X,-X)2 a) Derive the expected values of X and V b) Further suppose that Xi,...,Xn are normally distributed. Let Anxn - ((a) be an orthogonal matrix whose first row is (mVm Y = (y, . . . ,%), and X = (Xi, , Xn), are (column) vectors. (It is not necessary to know aij for...
Let X1, X2,.......Xn be a random sample of size n from a continuous distribution symmetric about . For testing H0: = 10 vs H1: < 10, consider the statistic T- = Ri+ (1-i), where i =1 if Xi>10 , 0 otherwise; and Ri+ is the rank of (Xi - 10) among |X1 -10|, |X2-10|......|Xn -10|. 1. Find the null mean and variance of T- . 2. Find the exact null distribution of T- for n=5. We were unable to transcribe this imageWe were...
3. Let X1, X2, . . . , Xn be random variables with a common mean μ. Sup- pose that cov[Xi, xj] = 0 for all i and A such that j > i+1. If 仁1 and 6 VECTORS OF RANDOM VARIABLES prove that = var X n(n- 3)
Let X1,X2, , Xn be a random sample from a normal distribution with a known mean μ (xi-A)2 and variance σ unknown. Let ơ-- Show that a (1-α) 100% confidence interval for σ2 is (nơ2/X2/2,n, nơ2A-a/2,n). Let X1,X2, , Xn be a random sample from a normal distribution with a known mean μ (xi-A)2 and variance σ unknown. Let ơ-- Show that a (1-α) 100% confidence interval for σ2 is (nơ2/X2/2,n, nơ2A-a/2,n).
Please answer question (a) X1 - X X2 – Å a. Let X1, ..., Xn i.i.d. random variables with X; ~ N(u, o). Express the vector in the | Xn – form AX and find its mean and variance covariance matrix. Show some typical elements of the vari- ance covariance matrix. b. Refer to question (a). The sample variance is given by S2 = n11 21–1(X; – X)2, which can be ex- pressed as S2 = n1X'(I – 111')X (why?)....
5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J, 1,,-1, , n. OV&.for any two random variables X and Y) or each 1, and (11 CoV(X,Y) var(x)var(y) (Recall that p vararo 5. Let X1,X2, . , Xn be a random sample from a distribution with finite variance. Show that (i) COV(Xi-X, X )-0 f ) ρ (Xi-XX,-X)--n-1, 1 # J,...
Let X1, X2, ..., Xn be a random sample of size 5 from a normal population with mean 0 and variance 1. Let X6 be another independent observation from the same population. What is the distribution of these random variables? i) 3X5 – X6+1 ii) W, = - X? iii) Uz = _1(X; - X5)2 iv) Wą +xz v) U. + x vi) V5Xe vii) 2X
Let X1, X2, ....,. Xn, be a set of independent random variables, each distributed as a normal random variable with parameters μί and σ. Let х, ai Use properties of moment generating functions to determine the distribution of Y, meaning: find the type of distribution we get, and its expected value and variance