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midterm2_practice1_solution

# midterm2_practice1_solution - Statistics 116 Fall 2004...

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Statistics 116 - Fall 2004 Theory of Probability Midterm # 2, Practice # 1 show (and briefly explain) all of your work. calculators are permitted for numerical calculations only. Instructions: Answer 4 out of 5 questions. Clearly mark which 4 questions you decide to answer. If you do not clearly indicate which 4 are to be counted, your mark will be based on 5 instead of 4 questions, there are no bonus points. All questions have equal weight. Q. 1) Let X be a Binomial random variable with parameters n and p . Show that E 1 X + 1 = 1 - (1 - p ) n +1 ( n + 1) p . Solution: E 1 X + 1 = n j =0 1 j + 1 n j p j (1 - p ) n - j = n j =0 n ! ( n - j )!( j + 1)! p j (1 - p ) n - j = 1 ( n + 1) p n j =0 ( n + 1)! ( n - j )!( j + 1)! p j +1 (1 - p ) n - j = 1 ( n + 1) p n +1 j =1 ( n + 1)! ( n + 1 - j )!( j )! p j (1 - p ) n +1 - j = 1 ( n + 1) p ( 1 - (1 - p ) n +1 ) 1

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Q. 2) A filling station is supplied with gasoline once a week. Suppose its weekly volume of sales in thousands of gallons is a random variable with proba- bility density function f ( x ) = 5(1 - x ) 4 0 < x < 1 0 otherwise.
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midterm2_practice1_solution - Statistics 116 Fall 2004...

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