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The spring 2000 midterm exam for the probability and random processes course (eecs126) at the university of california, berkeley. The exam includes four problem sheets, each with multiple questions, and covers topics such as probability calculations and probability distributions.
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Department of EECS - University of California at Berkeley EECS126 - Probability and Random Pro cesses - Spring 2000 Midterm No. 1: 2/23/
Name and SID:
Answer the questions on these four sheets. Show your work. Go o d luck.
Problem 1: (25%) You ip a fair coin rep eatedly. What is the probability that you have to ip it exactly 10 times to see two \heads"?
Problem 2: (25%) Let A; B ; C b e three events. Assume that P (A) = 0 : 6 ; P (B ) = 0 : 6 ; P (C ) = 0 : 7 ; P (A \ B ) = 0 : 3 ; P (A \ C ) = 0 : 4 ; P (B \ C ) = 0 : 4 ; P (A [ B [ C ) = 1 : Find P (A \ B \ C ):
Problem 4: (25%) De ne the random variable X as follows. You throw a dart uniformly in a circle with radius 5. The random variable X is equal to 2 minus the distance b etween the dart and the center of the circle if this distance is less than or equal to one. Otherwise, X is equal to 0. a. Plot carefully the probability distribution function F (x) = P (X x) for x 2 < := ( 1; + 1 ). b. Give the mathematical expression for the probability density function f (x) of X for x 2 < := ( 1; + 1 ).
