1 2 3 4 5 Bonus Total Question 1 Let fxx2 for x Î 02 a Does fx satisfy the

1 2 3 4 5 bonus total question 1 let fxx2 for x î 02

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1 2 3 4 5 Bonus Total
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Question 1 Let f(x)=x/2 for x Î (0,2). a) Does f(x) satisfy the necessary conditions to be a density function? Explain your answer. (5 points) b) What is the cumulative distribution function associated to f(x)? (5 points) c) What is the probability that X is between 0 and 1? (5 points)
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Question 2 Imagine that you have not studied for this midterm. There are 10 multiple-choice questions and for each of these questions there are 4 possible choices (for example, the correct answer is a, b, c, or d). You decide to answer these 10 questions in a purely random way, assuming that each of the options (a, b, c, and d) has the same probability of being the correct answer. a) What is the probability of having 3 good answers out of 10? (6 points) b) What grade should you expect if you answer these 10 questions randomly assuming you have two points per correct answer and 0 per wrong answer? (6 points) c) What would be the variance of the score if you answer these 10 questions randomly, assuming you have two points per correct answer and 0 per wrong answer? (6 points)
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Question 3 a) You receive three playing cards (from a pack of 52). What is the probability that you receive two cards of the same rank? Remember that there are 13 ranks (ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen and king) and four suits (clubs, spades, hearts and tiles) in a card game. (7 points) b) Let A i (i=1,…,n) be n mutually exclusive and collectively exhaustive events and B j another event. Show that (8 points)
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c)
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