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Problem 5.1 (Relation between Gaussian and exponential) Suppose that Xi and X, are i.i.d. N(0,1) (a)...
Answer all parts please. Question 3: Suppose that X and Y are i.i.d. N(0,1) r.v.'s (a) (5 marks). Find the joint pdf for U Х+Ү andV — X+2Y. (b) (3 marks). The joint pdf of U and V is for what particular distribution? (Hint: See p.81 of the textbook.) (c) (2 marks). Are U and V independent? Why?
2. The random variables X1, X2 and X3 are independent, with Xi N(0,1), X2 N(1,4) and X3 ~ N(-1.2). Consider the random column vector X-Xi, X2,X3]T. (a) Write X in the form where Z is a vector of iid standard normal random variables, μ is a 3x vector, and B is a 3 × 3 matrix. (b) What is the covariance matrix of X? (c) Determine the expectation of Yi = Xi + X3. (d) Determine the distribution of Y2...
4. Suppose X1, . . . ,X, are independent, normally distributed with mean E(Xi) and variance Var(X)-σί. Let Żi-(X,-μ.)/oi so that Zi , . . . , Ζ,, are independent and each has a N(0, 1) distribution. Show that LZhas a x2 distribution. Hint: Use the fact that each Z has a xî distribution i naS
Problem 4 True or False A Bookmark this page Instructions: Be very careful with the multiple choice questions below. Some are "choose all that apply," and many tests your knowledge of when particular statements apply As in the rest of this exam, only your last submission will count. 1 point possible (graded, results hidden) The likelihood ratio test is used to obtain a test with non-asymptotic level o True O False Submit You have used 0 of 3 attempts Save...
1 Expectation, Co-variance and Independence [25pts] Suppose X, Y and Z are three different random variables. Let X obeys Bernouli Distribution. The probability disbribution function is 0.5 x=1 0.5 x=-1 Let Y obeys the standard Normal (Gaussian) distribution, which can be written as Y are independent. Meanwhile, let Z = XY. N(0,1). X and Y (a) What is the Expectation (mean value) of X? 3pts (b) Are Y and Z independent? (Just clarify, do not need to prove) [2pts c)...
4. Setup: Suppose you have observations X1,X2,X3,X4,X5 which are i.i.d. draws from a Gaussian distribution with unknown mean μ and unknown variance σ2. Given Facts: You are given the following: 15∑i=15Xi=0.90,15∑i=15X2i=1.31 Bookmark this page Setup: Suppose you have observations X1, X2, X3, X4, X5 which are i.i.d. draws from a Gaussian distribution with unknown mean u and unknown variance o? Given Facts: You are given the following: x=030, =1:1 Choose a test 1 point possible (graded, results hidden) To test...
As on the previous page, let Xi,...,Xn be i.i.d. with pdf where >0 2 points possible (graded, results hidden) Assume we do not actually get to observe X, . . . , Xn. to estimate based on this new data. Instead let Yİ , . . . , Y, be our observations where Yi-l (X·S 0.5) . our goals What distribution does Yi follow? First, choose the type of the distribution: Bernoulli Poisson Norma Exponential Second, enter the parameter of...
X Part I. Derive Bivariate Regression by hand. Again, we are using the same data set that we used in the in-class assessment. Case Dietary Fat Body Fat 22 9.8 22 11.7 14 8.0 21 9.7 32 10.9 26 7.8 30 21 17 1. Step 1: Find the mean of dietary fat x = 2. Step 2: Find the mean of body fat y = 3. Step 3: Find the sum of (x1 - x)y- y) = 3316 4. Step...