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Homework due 09/15 1. Consider a sequence of five Bernoulli trials. Let X be the number of times that a head is followed immediately by a tail. For example, if the outcome is ω = HHT HT then X(ω) = 2 since a head is followed directly by a tail at trials 2 and 3, and also at trials 4 and 5. Find the probability mass function of X, assume that p = 12 . 2. We roll a fair die three times. Let X be the number of times that we roll a 6. What is the probability mass function of X? 3. Some day, 10,000 cars are travelling across a city. Suppose that the probability that a car has an accident this day is 0.002. Using the approximation of a binomial distribution by a Poisson distribution, compute the probability that exactly 15 cars have an accident this day. 4. Let F be the function defined by: F (x) = 0 2 x3 1 3 1 x+ 6 1 1 3 x<0 0≤x<1 1≤x<2 2≤x<4 4≤x Let X be a random variable which corresponds to F . 1. Verify that F is a cumulative distribution function. (You need to verify the conditions (a)-(c) we mentioned in class.) 2. Compute P (X = 2). 3. Compute P (X < 2). 4. Compute P (X = 2 or 1 2 ≤ X < 32 ). 5. Compute P (X = 2 or 1 2 ≤ X ≤ 3). 1