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2010100AHW2 - state 1 Each time he stays where he is with...

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STAT 100A HWII Note (1) Rule of total probability. Suppose B 1 , B 2 , ..., B n partition the sample space, then P ( A ) = n i =1 P ( A B i ) = n i =1 P ( B i ) P ( A | B i ). (2) Bayes rule. P ( B j | A ) = P ( B j A ) /P ( A ) = P ( B j ) P ( A | B j ) / n i =1 P ( B i ) P ( A | B i ). Please show all the necessary steps in your calculations. Please be precise with notation. Problem 1: Suppose an urn has b blue balls and r red balls. We randomly pick a ball. If the ball is red, we put two red balls back to the urn. If the ball is blue, we put two blue balls back to the urn. Then we randomly pick a ball again. (1) What is the probability that the first pick is red? (2) What is the probability that the second pick is red? (3) (Optional) If we continue, then what is the probability that the third pick is red? Problem 2: Suppose a person performs a random walk over three states, 1, 2, 3. He starts from
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Unformatted text preview: state 1. Each time, he stays where he is with probability 1/2, and he moves to each one of the other two states with probability 1/4. Let X t be the state of this person at time t , and X = 1. (1) What is the distribution of X 1 , i.e., what is the probability that X 1 = k for k = 1 , 2 , 3? (2) What is the distribution of X 2 ? (3) (Optional) What is the distribution of X 3 ? Problem 3: Suppose at any moment, the probability that there is fire in a classroom is α . If there is a fire, the probability we hear the fire alarm is β . If there is not a fire, the probability that there is fire alarm is γ . Given that we hear the fire alarm, what is the probability that there is fire? 1...
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