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Department of Electrical Engineering and Computer Sciences
EECS 126: Probability and Random Processes
Discussion 9
1. Jump Chain Stationary Distribution
Use properties of transient states and the jump chain to find the stationary distribution of this CTMC.

2. Two-Server System
A company has two servers (the second server is a backup in case the first server fails, so only one server is ever used at a time). When a server is running, the time until it breaks down is exponentially distributed with rate µ. When a server is broken, it is taken to the repair shop. The repair shop can only fix one server at a time, and its repair time is exponentially distributed with rate λ. Find the long-run probability that no servers are operational.
3. Gaussians and the MSE
Suppose you draw n i.i.d. data points (x1,y1),…,(xn,yn), where n is a positive integer and the true relationship is Y = WX + ε, ε ∼N(0,σ2). (That is, Y has a linear dependence on X, with additive Gaussian noise.) Show that finding the MLE estimate of W given the data points {(xi,yi) : i = 1,…,n} is equivalent to minimizing the cost function

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