180afinalrev - MATH 180A Introduction to Probability Review...

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MATH 180A – Introduction to Probability Review for the Final (NOT due) Please review homework sets 1-7, midterms 1 & 2, midterm reviews 1 & 2. The following problems focus on what was covered since midterm 2. 1. Two balls are drawn with replacement from an urn containing n balls numbered 1 through n . Let X be the number the first draw and Y the number on the second draw. Compute the PDF of Z = X + Y . Repeat when the balls are drawn without replacement. 2. A fair die is rolled repeatedly. Let X be the number of trials until all numbers 1 through 6 appear. For example, X = 11 in the following example: 1 4 2 2 5 6 2 1 1 4 3 2 2 2 5 4 1 4 4 Compute E ( X ). (Hint: Consider X i , the number of trials required to get a new number given that i - 1 different numbers have already appeared. Express X in terms of the X i ’s and show that each X i has a geometric distribution.) 3. Consider X and Y with the following joint PDF: f ( x, y ) = C (1 - x - y ) , x, y (0 , 1) , x + y 1 . (a) Compute C . (b) Compute E ( X ), var( X ), E ( Y ), var( Y ). (c) Compute E ( X + Y ) and var( X + Y ). 4. Consider X , Y and Z with the following joint PDF: f ( x, y, z ) = C ( z - x )( z - y ) , 0 < x < z, 0 < y < z, 0 < z < 1 . (a) Compute C . (b) Compute E ( X ), E ( Y ), E ( Z ). (c) Compute the marginals
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  • Fall '08
  • Castro
  • Sets, Probability, Probability theory, Exponential distribution, ProbabiliTy Review, Compute C.

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