rec11-1 - etc are modeled well with Poisson random...

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Massachusetts Institute of Technology 6.041/6.431: Probabilistic Systems Analysis (Fall 2008) Recitation 11 October 9, 2008 1. Let X be a continuous random variable with the uniform distribution on [ - 1 , 1]. Find the PDF of r | X | and the PDF of - ln | X | . 2. (Problem 3.34, page 198 of the text.) A defective coin minting machine produces coins whose probability of heads is a random variable X with PDF f X ( x ) = b xe x , x [0 , 1] , 0 , otherwise . A coin produced by this machine is selected and tossed repeatedly, with successive tosses assumed independent. (a) Find the probability that a coin toss results in heads. (b) Given that a coin toss resulted in heads, ±nd the conditional PDF of X . (c) Given that the ±rst coin toss resulted in heads, ±nd the conditional probability of heads on the next toss. 3. Many natural phenomena involving counts (emitted or absorbed photons, decaying particles,
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Unformatted text preview: etc.) are modeled well with Poisson random variables. Suppose that for each minute that a Geiger counter is exposed to some source of radiation, the number of “beeps” N has the Poisson PMF with parameter λ > 0: p N ( n ) = e-λ λ n n ! , for n = 0 , 1 , 2 , ... (and zero otherwise) . The numbers of beeps in disjoint minutes are independent. (a) Find the probability of observing 1, 3, 3, and 2 beeps in four disjoint minutes. (That is, the probability of observing 1 beep in the ±rst minute, 2 beeps in the second minute, and so on.) The answer will depend on λ . (b) As a way of estimating λ from the observations in part (a), ±nd the value of λ that maximizes the probability found in part (a). This estimate of λ is called the maximum likelihood estimate. We will return to this topic in Chapter 9. Compiled October 7, 2008 Page 1 of 1...
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