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homework2 - Homework 2 36-325/725 due Friday Sept 7 Hint...

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Homework 2 36-325/725 due Friday Sept 7 (1) Chapter 1.10 Problem # 11. Let be an infinite sequence of events such that . Show that Hint: Define , , , etc. Next show that (i) the are disjoint, (ii) and (iii) for every . Also, it will help to recall the definition of an infinite sum: . (2) Chapter 1.11 Problem # 1 Suppose that and are independent events. Show that and are inde- pendent events. (3) Chapter 1.13 Problem # 1 Suppose that a fair coin is tossed repeatedly until both a head and tail have appeared at least once. (a) Describe the sample space . (b) What is the probability that three tosses will be required? (4a) Show that if or then is independent of every other event. (4b) Show that if is independent of itself then is either 0 or 1. (5) Suppose a coin has probability of falling heads. If we flip the coin many times, we would expect the proportion of heads to be near . We will make this formal later. Let’s explore the idea now using R (or Splus). Let’s pick a value of and generate coin flips: p <- .3 n <- 1000 1
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### generate n coin flips each having prob p
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