Please give the detailed explanation, Thanks!! Denote the sample correlation between x1,...xn and y1,...yn as rx,y. Let wi=a+bi and let zi=c+dyi for i=1,...,n where b>0 and d>0. What is the relationship between rx,y and rw,z?
Please give the detailed explanation, Thanks!! Denote the sample correlation between x1,...xn and y1,...yn as rx,y....
Consider a random sample (X1, Y1),(X2, Y2), . . . ,(Xn, Yn) where Y | X = x is modeled by a N(β0 + βx, σ2 ) distribution, where β0, β1 and σ 2 are unknown. (a) Prove that the mle of β1 is an unbiased estimator of β1. (b) Prove that the mle of β0 is an unbiased estimator of β0.
1. Let X1, ..., Xn, Y1, ..., Yn be mutually independent random variables, and Z = + Li-i XiYi. Suppose for each i E {1,...,n}, X; ~ Bernoulli(p), Y; ~ Binomial(n,p). What is Var[Z]?
Exercise 3.8.5 (Cosine and sample correlation) Given two samples of observa tions Y = (y1, ,yn) and X = (zi, ,xn). Y = yin and X = iln. Le ,2= 4, i) Express the sample correlation between X andY in tems of is, yis, i and y. ii) Express the cosine of the angle between Y -Y and X - X in terms of r,s, yis, and y iii) Sketch a subject space plot to represent Y, X, Y, x,...
. Suppose the Y1, Y2, · · · , Yn denote a random sample from a population with Rayleigh distribution (Weibull distribution with parameters 2, θ) with density function f(y|θ) = 2y θ e −y 2/θ, θ > 0, y > 0 Consider the estimators ˆθ1 = Y(1) = min{Y1, Y2, · · · , Yn}, and ˆθ2 = 1 n Xn i=1 Y 2 i . ii) (10 points) Determine if ˆθ1 and ˆθ2 are unbiased estimators, and in...
QUESTION 3 Let Y1, Y2, ..., Yn denote a random sample of size n from a population whose density is given by (Parcto distribution). Consider the estimator β-Yu)-min(n, Y, where β is unknown (a) Derive the bias of the estimator β. (b) Derive the mean square error of B. , Yn).
Please give detailed steps. Thank you. 5. Let {X1, X2,..., Xn) denote a random sample of size N from a population d escribed by a random variable X. Let's denote the population mean of X by E(X) - u and its variance by Consider the following four estimators of the population mean μ : 3 (this is an example of an average using only part of the sample the last 3 observations) (this is an example of a weighted average)...
8. Let X1,...,Xn denote a random sample of size n from an exponential distribution with density function given by, 1 -x/0 -e fx(x) MSE(1). Hint: What is the (a) Show that distribution of Y/1)? nY1 is an unbiased estimator for 0 and find (b) Show that 02 = Yn is an unbiased estimator for 0 and find MSE(O2). (c) Find the efficiency of 01 relative to 02. Which estimate is "better" (i.e. more efficient)? 8. Let X1,...,Xn denote a random...
Let's say (XL, Y)',.., (Xn,Yn)'(n > 2) is a random sample at Bivariate Normal Distribution ρσ102 41(ΑΓΑΝ! and r is sample correlation coefficient r- Also when Z and W, is as below, answer following questions (Question 3) (1) Show independence between (Z1, ..,Zn)' and (W, W)' (2) When (,,..,Zn)' and (Wi.,W' is independent, show Sww- Zw ~x2(n - 2), and show it is Szz ,Zn) independent with (Z1 Szw is independent, showNO,.1), and show it is 3) When (Z Z)...
Let X1, X2, ..., Xn denote a random sample of size n from a population whose density fucntion is given by 383x-4 f S x f(x) = 0 elsewhere where ß > 0 is unknown. Consider the estimator ß = min(X1, X2, ...,Xn). Derive the bias of the estimator ß.
Please give some detailed explanation on this question. Thanks! 2.12 The t-test for slope as a function of the correlation Show that the t-statistic for testing the slope = 0 can be written as a function of sample size and the sample correlation rry, (2.26)