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1. Let {Xn}n≥0 be a discrete-time Markov chain on the state space S = {1,2,3} with transition matrix P =  1/3 1/3 1/3 0 2/3 1/3 1/3 2/3 0  .If the initial distribution is
P(X0 = 1) = P(X0 = 2) = P(X0 = 3) = 1/3, what is P(X3 ∈{2,3})? 2. Let Π be a homogeneous Poisson point process on the line, with intensity λ. Let r+ be the distance from the origin O to the closest point of Π on the positive axis and r− be the distance from O to the closest point of Π on the negative axis. Find the distribution functions of these random variables: i.e., find P(r+ ≤ x) and P(r− ≤ x) for x ≥ 0. Find also the distribution function of the distance from O to the closest point of Π: i.e., find P(min(r−,r+) ≤ x). 3. Let the continuous-time Markov process {X(t)}t≥0 on the state space S = {1,2,3,4} have the matrix of transition rates

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Tellus molestie nunc non blandit massa enim nec dui. Tellus molestie nunc non blandit massa enim nec dui. Ac tortor vitae purus faucibus ornare suspendisse sed nisi. Pharetra et ultrices neque ornare aenean euismod. Pretium viverra suspendisse potenti nullam ac tortor vitae. Morbi quis commodo odio aenean sed. At consectetur lorem donec massa sapien faucibus et. Nisi quis eleifend quam adipiscing vitae proin sagittis nisl rhoncus. Duis at tellus at urna condimentum mattis pellentesque. Vivamus at augue eget arcu dictum varius duis at. Justo donec enim diam vulputate ut. Blandit libero volutpat sed cras ornare arcu. Ac felis donec et odio pellentesque diam volutpat commodo. Convallis a cras semper auctor neque. Tempus iaculis urna id volutpat lacus. Tortor consequat id porta nibh.

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Tellus molestie nunc non blandit massa enim nec dui. Tellus molestie nunc non blandit massa enim nec dui. Ac tortor vitae purus faucibus ornare suspendisse sed nisi. Pharetra et ultrices neque ornare aenean euismod. Pretium viverra suspendisse potenti nullam ac tortor vitae. Morbi quis commodo odio aenean sed. At consectetur lorem donec massa sapien faucibus et. Nisi quis eleifend quam adipiscing vitae proin sagittis nisl rhoncus. Duis at tellus at urna condimentum mattis pellentesque. Vivamus at augue eget arcu dictum varius duis at. Justo donec enim diam vulputate ut. Blandit libero volutpat sed cras ornare arcu. Ac felis donec et odio pellentesque diam volutpat commodo. Convallis a cras semper auctor neque. Tempus iaculis urna id volutpat lacus. Tortor consequat id porta nibh.

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MFA501 Mathematical Foundations of Artificial Intelligence

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