MATH 180A – Introduction to Probability Review for the Final (NOT due) 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
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