Winter 2006 Review Session Solutions

Winter 2006 Review Session Solutions - Some Selected...

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Some Selected Solutions These are the solutions to those questions I have been asked in preparation for the exam. 7.7 (Notes) Suppose that n people take a blood test for a disease, where each person has probability p of having the disease, independent of other persons. To save time and money, blood samples from k people are pooled and analyzed together. If none of the k persons has the disease then the test will be negative, but otherwise it will be positive. If the pooled test is positive then each of the k persons is tested separately (so k +1 tests are done in that case). (a) Let X be the number of tests required for a group of k people. Show that E ( X )= k k (1 p ) k . (b) What is the expected number of tests required for n/k groups of k people each? If p = . 01 , evaluate this for the cases k =1 , 5 , 10 . (c) Show that if p is small, the expected number of tests in part (b) is approximately n ( kp + k 1 ) ,andism in im izedfor k . = p 1 / 2 . (b) The number of tests required for n/k groups of k people each is n k E ( X n k ( k k (1 p ) k ) Do part (c): When p is small, (1 p ) k + μ k 2 ( p ) 2 + .... ' 1 Substituting, n k E ( X n k [ k k (1 p ) k ] ' n k [ k k (1 )] = n k [1 + k 2 p ]= n [ 1 k + ] Minimizing this over k as if k were a continuous variable, taking the derivative with respect to k and setting it equal to 0 gives k = p 1 / 2 1
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8.8 (Notes) In a quality control inspection items are classi f ed as having a minor defect, a major defect, or as being acceptable. A carton of 10 items contains 2 with minor defects, 1 with a major defect, and 7 acceptable. Three items are chosen at random without replacement. Let X be the number selected with minor defects and Y be the number with major defects. a) Find the joint probability function of X and Y . b) Find the marginal probability functions of X and of Y . c) Evaluate numerically P ( X = Y ) and P ( X =1 | Y =0) . Denote the possible outcomes M (major), m (minor), A (acceptable). We take a sample of size 3 from the set { M, m, m, A, A, A, A, A, A, A } . Then f ( x, 0) = P ( X = x, Y =0)= ¡ 2 x ¢¡ 7 3 x ¢ ¡ 10 3 ¢ ,x =0 , 1 , 2 f ( x, 1) = P ( X = x, Y =1)= ¡ 2 x ¢¡ 7 2 x ¢ ¡ 10 3 ¢ , 1 , 2 (b) P ( X = x )= ¡ 2 x ¢¡ 7 3 x ¢ + ¡ 2 x ¢¡ 7 2 x ¢ ¡ 10 3 ¢ , 1 , 2 (c) P ( X = Y f (0 , 0) + f (1 , 1) = ¡ 2 0 ¢¡ 7 3 ¢ ¡ 10 3 ¢ + ¡ 2 1 ¢¡ 7 2 1 ¢ ¡ 10 3 ¢ = ¡ 7 3 ¢ +14 ¡ 10 3 ¢ = 49 120 . 40833 P ( Y ¡ 9 3 ¢ ¡ 10 3 ¢ = 7 10 P ( X | Y f (1 , 0) P ( Y = ¡ 2 1 ¢¡ 7 3 1 ¢ ¡ 10 3 ¢ 7 10 = 1 2 2
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8.18 (Notes) A multiple choice exam has 100 questions, each with 5 possible answers. One mark is awarded for a correct answer and 1/4 mark is deducted for an incorrect answer. A particular student has probability p i of knowing the correct answer to the i th question, independently of other questions. a) Suppose that on a question where the student does not know the answer, he or she guesses randomly. Show that his or her total mark has mean P p i and variance P p i (1 p i )+ (100 S p i ) 4 .
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Winter 2006 Review Session Solutions - Some Selected...

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