Question

Thank you so much in advance

1. (10 points) (Without Python) Suppose that there are 4 songs on your professor's half marathorn running playlist (e.g. Eastside; Better Now; Lucid Dreams; Harder, Better, Faster, Stronger). She sets it in shuffle mode which plays songs uniformly at random, sampling with replacement (i.e. repeats are possible). Let Xn be the number of unique songs that have been heard after the nth song played with Xo -0 (a) Briefly explain why (Xn)20 forms a stationary discrete time Markov chain (b) Find the transition probability matrix. c) Suppose your professor has listened to 3 songs. Compute the probability that all the songs were unique. If you use the Markov property, please indicate where it is used. (d) After 3 songs have been played, what is the expected number of unique songs that your professor has listened to?

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Answer 1