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Unformatted text preview: IE 230 Seat # ________ Name _______________________ Closed book and notes. 60 minutes. Cover page and four pages of exam. Pages 8 and 12 of the Concise Notes. No calculator. No need to simplify answers. This test is cumulative, with emphasis on Section 4.7 through Chapter 6 of Montgomery and Runger, fourth edition. Remember: A statement is true only if it is always true. One point: On the cover page, circle your family name. One point: On every page, write your name. The random vector ( X 1 , X 2 , . . . , X k ) has a multinomial distribution with joint pmf P( X 1 = x 1 , X 2 = x 2 , . . . , X k = x k ) = x 1 ! x 2 ! . . . x k ! n ! hhhhhhhhhhhhh p 1 x 1 p 2 x 2 . . . p k x k when each x i is a nonnegative integer and x 1 + x 2 + . . . + x k = n ; zero elsewhere. The linear combination Y = c + c 1 X 1 + c 2 X 2 + . . . + c n X n has mean and variance E( Y ) = c + i = 1 Σ n E( c i X i ) = c + i = 1 Σ n c i E( X i ) and V( Y ) = i = 1 Σ n j = 1 Σ n cov( c i X i , c j X j ) = i = 1 Σ n j = 1 Σ n c i c j cov( X i , X j ) = i = 1 Σ n c i 2 V ( X i ) + 2 i = 1 Σ n − 1 j = i + 1 Σ n c i c j cov( X i , X j ) . Cov( X , Y ) = E[( X − μ X ) ( Y − μ Y )] Corr( X , Y ) = Cov( X , Y ) / ( σ X σ Y ) X d = Σ i = 1 n X i / n S 2 = Σ i = 1 n ( X i − X d ) 2 / ( n − 1) Order statistics satisfy X (1) ≤ X (2) ≤ . . . ≤ X ( n ) . Score ___________________________ Exam #3, Nov 30, 2010 Schmeiser IE 230 Seat # ________ Name _______________________ Closed book and notes. 60 minutes. 1. (3 points each) True or false. Consider two continuous random variables X and Y with probability density functions f X and f Y , expected values μ X and μ Y , standard deviations σ X and σ Y , and correlation ρ X , Y . (a) T F μ X ( v ) = ∫ −∞ ∞ f X ( v ) dv (b) T F f X , Y ( x , y ) = f Y ( y ) f X | Y = y ( x ) (c) T F E( X − Y ) = μ X − μ Y (d) T F Var( X − Y ) = σ X 2 − σ Y 2 (e) T F | Cov( X , Y ) | ≤ σ X σ Y (f) T F If ρ X , Y = 0, then X and Y are independent. (g) T F If X and Y are independent, then ρ X , Y = 0. (h) T F If ( X , Y ) is bivariate normal, then both X and Y are normal. 2. (3 points each) Consider the notation from Question 1 above. For each of the following, indicate whether the expression is a constant, an event, a random variable, or undefined. (A constant has the same numerical value for every replication of the experiment.) (a) σ X σ Y constant event random variable undefined (b) μ X X constant event random variable undefined (c) X 1 / 2 / Y 1 / 2 constant event random variable undefined (d) X > 1 / 2 constant event random variable undefined (e) XY > σ X σ Y constant event random variable undefined Exam #3, Nov 30, 2010 Page 1 of 4 Schmeiser IE 230 — Probability & Statistics in Engineering I...
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- Fall '08
- Normal Distribution, Probability theory, Cumulative distribution function, event event event