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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 4.8 through Section 5.5 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 n E( c i X i ) = c + i = 1 n c i E( X i ) and V( Y ) = i = 1 n j = 1 n cov( c i X i , c j X j ) = i = 1 n j = 1 n c i c j cov( X i , X j ) = i = 1 n c i 2 V ( X i ) + 2 i = 1 n 1 j = i + 1 n c i c j cov( X i , X j ) . Cov( X , Y ) = E[( X X ) ( Y Y )] Corr( X , Y ) = Cov( X , Y ) / ( X Y ) If ( X , Y ) is bivariate normal, then X and Y are normal. In addition, the conditional distribution of X given that Y = y is normal, with mean X + X , Y X [( y Y ) / Y ] and variance (1 X , Y 2 ) X 2 . Score ___________________________ Exam #3, April 12, 2011 Schmeiser IE 230 Probability & Statistics in Engineering I Name _______________________ Closed book and notes. 60 minutes. 1. (3 points each) True or false. Consider the notes from the cover page. (a) T F E( X Y ) = X + Y . (b) T F Corr( X , Y ) 0 implies that Cov( X , Y ) 0. (c) T F If ( X , Y ) is bivariate normal, then X , Y = 0. (d) T F If X 1 and X 2 are independent and exponentially distributed, then X 1 + X 2 has an exponential distribution. (e) T F If ( X , Y ) is bivariate normal, then 4.5( X + Y ) is normally distributed. (f) T F If v and w are real numbers, then f X , Y ( v , w ) = f X ( v ) f Y ( w ). (g) T F If X and Y are independent, then X , Y = 0. (h) T F The time to recharge a battery is never normally distributed. 2. (3 points each) Consider the notation from the cover page. For each of the following, indicate whether the expression is a constant, an event, a random variable, or undefined....
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- Spring '08