Midterm1

# Midterm1 - Stat 5101 First Midterm Exam Name Student ID 1...

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Stat 5101 First Midterm Exam October 14, 2009 Name Student ID The exam is closed book and closed notes. You may use one 8 1 2 × 11 sheet of paper with formulas, etc. You may also use the handout on “brand name distributions”. Put all of your work on this test form (use the back if necessary). Show your work or give an explanation of your answer. No credit for numbers with no indication of where they came from. The points for the questions total to 100. There are 6 pages and 5 prob- lems. 1. [20 pts.] Suppose X is a random variable having probability mass function (PMF) given by x 1 2 3 4 5 f ( x ) 1 / 7 2 / 7 1 / 7 2 / 7 1 / 7 (a) Calculate E ( X ). (b) Calculate var( X ). 1

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(c) Calculate Pr( X > 3). 2. [20 pts.] Suppose X is a random variable having PMF given by x 1 2 3 4 5 6 7 f ( x ) 1 / 28 1 / 14 3 / 28 1 / 7 5 / 28 3 / 14 1 / 4 Find the PMF of the random variable Y = ( X - 2)( X - 4)( X - 6). 2
3. [20 pts.] Suppose X is a random vector having mean vector μ = 1 2 3 and variance matrix M = 1 0 1 0 2 1 1 1 3 And suppose B = 1 1 1 3 2 1 and define the random vector Y = BX (the right hand side being a matrix multiplication). (a) Calculate the mean vector of Y . (b) Calculate the variance matrix of Y . 3

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4. [20 pts.] Suppose the random vector (

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