# hw3_2011 - t If X ≤ t Alvin gets a score of S = 1 X...

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Theory of Probability, HW #3, all sections Assignment Date: April 8, 2011 Due Date: April 15, 2011 1. An absent-minded professor schedules two student appointments for the same time. The appointment durations are independent and exponentially distributed with mean thirty minutes. The first student arrives on time, but the second student arrives five minutes late. What is the expected value of the time between the arrival of the first student and the departure of the second student? 2. Alvin throws darts at a circular target of radius r and is equally likely to hit any point in the target. Let X be the distance of Alvin’s hit from the center. a) Find the pdf, the mean, and the variance of X . b) The target has an inner circle of radius
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Unformatted text preview: t . If X ≤ t , Alvin gets a score of S = 1/ X . Otherwise his score is S = 0. Find the cdf of S . Is S a continuous random variable? 3. Let X be a uniform (0, 1) random variable. Compute g[G ± ] and then check the result by using the definition of expectation. 4. Let X be uniformly distributed on the interval (0, 1). Find the density function of ² = G ³/´ where α ≠ 0. 5. The lung cancer hazard rate of a t-year-old male smoker, λ ( t ), is such that µ¶·¸ = .027 + .00025¶· − 40¸ ¹ · ≥ 40 Assuming that a 40-year-old male smoker survives all other hazards, what is the probability that he survives to a) age 50 and, b) age 60 without contracting lung cancer?...
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