STAT 8320 Homework 1: Probability Distributions and Hypothesis Testing, Assignments of Statistics

Download STAT 8320 Homework 1: Probability Distributions and Hypothesis Testing and more Assignments Statistics in PDF only on Docsity! Homework 1 STAT 8320 Due February 7, 2007 Problem 1: Suppose that y ∼ N4(µ, Σ) where µ =   1 −3 5 −8   and Σ =   4 0 0 −6 0 16 8 0 0 8 9 0 −6 0 0 25   . a. Find the distribution of y3. b. Find the distribution of 2y1 − 5y2 + 3y3. c. Find the joint distribution of y2 and 3y1 + y2 − 2y3. d. Find the conditional distribution of y4 given y1 + y4 = −4. e. Which of the following pairs are independent: i. y1 and y2 − y3, ii. (y1, y3)′ and (y2, y4)′, iii. y1 + 3y2 and y3 + 4y4. Problem 2: A study about the amount of vitamins in a cup of spinach measures the following variables: y1: Vitamin A (mg) y2: Vitamin K (mcg) y3: Calcium (mg) y4: Folate (mcg) y5: Potassium (mg) Suppose that 30 samples are analyzed and give the following results: ȳ =   517.3 144.8 30.1 36.1 345.7   and S =   27.0 4.5 3.8 −1.4 −1.5 4.5 12.3 −0.4 −0.9 −0.8 3.8 −0.4 5.2 1.3 −1.4 −1.4 −0.9 1.3 5.3 0.0 −1.5 −0.8 −1.4 0.0 15.3   . 1