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Unformatted text preview: Introduction to Stochastic Processes, Spring 2011 Homework #2 (updated January 18th) Due: Tuesday, January 25th, 2011 in class 1. There are n urns, and the k th urn contains k- 1 red balls and n- k + 1 white balls. You pick an urn at random and remove two balls at random without replacement. Find the probability that: (a) the second ball is white, (b) the second ball is white, given that the first ball is white. 2. The exponential distribution with parameter λ > 0 has the probability density function f ( x ) = λe- λx , x ≥ , x < (a) Find the mean and variance of this distribution. (b) Find the moment generating function of this distribution, i.e. let X be a random variable with the exponential distribution and compute m ( t ) = E e tX . (c) For what values of t ∈ R is m ( t ) finite? (d) Use m ( t ) to compute the mean and variance of the exponential distribution. (e) Show that the exponential distribution has the memoryless property , that is show that for t,s ≥ 0 the relation P ( X > t + s | X > t ) = P ( X > s ) is true. Suppose that X is the lifespan of a light bulb. Use one sentence to summarize what the memoryless property says about the life of the light bulb. Remark. It can actually be shown that the exponential distribution is essentially the only probability distribution with the memoryless property. For this reason it plays an important role in the study of continuous-time Markov processes, as we shall see later. 1 3. The Monty Hall Problem. In the old game show Let’s Make a Deal , the contestant is presented with three doors and must pick one. Behind one door is a real prize and behind the other two is a booby prize. After the contestant chooses a door, the host of the show, Monty Hall (born in Winnipeg, BTW), reveals what is behind one of the remaining two doors and gives the contestant the option of switching their choice to the remaining door. Should the contestant switch or not? If they do switch, what isthe remaining door....
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- Spring '11
- Normal Distribution, Probability theory, Exponential distribution, yk, Red balls