explan the answer . Suppose that Xi, X2,.... Xn are independent random variables. Assume that E[A]-: μί ald Var(Xi)-σ? where i-| , 2, , n. If ai, aam., an are constants. (i) Write down expression for (i) E{E:-aiX.) and (ii) Var(Σ-lai%). (i) Rewrite the expression if X,'s are not independent.
Let {an} m-o and {bn}ņ=be any two sequences of real numbers, we define the following: N • For any real number L € R, we write an = L if and only if lim Lan = L. N-0 n=0 n=0 X • We write an = bn if and only if there is a real number L such that n=0 n=0 I and Σ. = L. Select all the correct sentences in the following list: X η (Α) Σ Σ...
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,...
4. Suppose Σί, 1 Xi-1, ΣΙ-, x-2, and n-5. Evaluate the followings: a) Σί-1[Xi + X) b) 5. Let X-1 Σ., x1-1 with n-100. (i) Obtain ΣΙ.i Xu (ii) Evaluate Σ.1x,-X) and X +1
theorem1 let an and bn be squences of real numbers theorem 2 let an and bn and cn be squences of real numbers if an<bn<cn theorem 3 let an be squences of real numbers if an=L and L defined at all an,f(an)=f(L) theorem 4 f(x) defined for all x>n0 then limit f(x)=L and limit an =L theorem 5 follwing six squences converage to be limit limit lnn\n =0 ,limit (1+x/n)n=ex .... Based on Theorems 1 to 5 in Section 10.1...
Q4 20 Points Let (a.) 21 be a sequence of real numbers and a ER such that .-+ 4. No fles uploaded Q4.1 10 Points State the definition of " a Please select flies a ". Select files Q4.2 5 Points in 2020. Consider the sequence (6.) 1 given by bn = 24 in <2020 and be Using only the definition of convergence of sequences, show b a . Please select file Select file Q4.3 5 Points Let(). be a...
4. Let X1,X2, x 2) distribution, and let sr_ Ση:1 (Xi-X)2 and S2 n-l Σηι (Xi-X)2 be the estimators of σ2. (i) Show that the MSE of S" is smaller than the MSE of S2 (ii) Find ElvS2] and suggest an unbiased estimator of σ. n be a random sample from N (μ, σ
an+1 for all values of n. What 1. Let {an} be a sequence of positive, real numbers such that is lim an? Explain how you got your answer. an 3n + 1 n-> 2. Let {an} and {bn} each be a sequence of positive real numbers. You know that ) bn converges and k=1 21. Your buddy Ron concludes that the series converges also. Select an item below and n70 bm 10. explain. lim An _ 1001
Let xi, i 1, 2, 3, , be a sequence of nonnegative numbers such that Σ x.-1 and consider the random variable X whose probability function is defined by: x, for x=x1, x2, X3, 0, for all other x What is the variance of X? i= 1
(a) If var[X o2 for each Xi (i = 1,... ,n), find the variance of X = ( Xi)/n. (b) Let the continuous random variable Y have the moment generating function My (t) i. Show that the moment generating function of Z = aY b is e*My(at) for non-zero constants a and b ii. Use the result to write down the moment generating function of W 1- 2X if X Gamma(a, B) (a) If var[X o2 for each Xi (i...