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160a06_mdrev

# 160a06_mdrev - /k When a key is put in a box that already...

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Pstat160a Winter 2006 Midterm Review Problem 1. (30 points) In this problem, we want to simulate the random variable with pdf given by p X (0) = . 2 , p X (1) = . 4 , p X (2) = . 3 , p X (3) = . 1 1) Recall that the inverse transformation method was the basic algorithm we saw in class. Describe it in the context of this example. 2) You are given the following 3 pseudo random numbers: .1234, .7656, .2343. Use them to generate 3 random variables with the pdf given above using the method from part 1). Problem 2. John and Mary play a game. Each of them roll a die until a six shows up. The one that get a six in the least number of throw wins. Let X 1 be the score of John and X 2 Mary’s. a) What is the distribution (pdf) of X 1 and X 2 ? b) What is the probability that they tie? (Hint: you may want write this in terms of the X ’s and use conditioning) c) What is the probability that John wins?

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2 Problem 3. (40 points) Suppose that we are given k boxes and r keys to be put at random one after another into these k boxes. Each box is equaly likely to recieve a key (so the probability to put a key in any box is 1
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Unformatted text preview: /k ). When a key is put in a box that already contains a key, we say that a collision occurs. The goal is to ﬁnd the expected number of such collisions X . We also let Y be the number of boxes with at least one key (at the end of the allocation process). For instance, if we have 3 boxes and 4 keys, with no keys in the ﬁrst box, 4 in the second and 1 in the third, X is 3 (3 keys were added to the ﬁrst one in box 2) and Y is 2. The problem is divided in several elementary steps, and you should not be surprised to ﬁnd strong similarity with one of the homework problems: 1) What is the probability that a given box will recieve no keys at all? At least one? 2) Using the familiar trick of deﬁning random variables with values 0 or 1, ﬁnd the expected value of Y . 3) Find a simple relationship between X and Y , and use it to compute the expected value of X (Hint: 3=5-2 in our example)...
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160a06_mdrev - /k When a key is put in a box that already...

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