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 > 0: p N ( n ) = e- 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 . (b) As a way of estimating from the observations in part (a), nd the value of that maximizes the probability found in part (a). This estimate of 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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