Task 3: Transformed Random Variables
A type of network router has a bandwidth total to first hardware failure called S expressed in terabytes. The random variable S is modelled by a distribution whose density is given by one of the following functions:
with a single parameter θ. Consider the bandwidth total to failure T of the sequence of the two routers of the same type (one being brought up automatically when the first is broken).
Express T in terms of the bandwidth total to failure of single routers S1 and S2. Formulate realistic assumptions about these random variables. Calculate the density function of the variable T.
Given an experiment with the dual-router-system yielding a sample T1, T2, …, Tn, calculate the likelihood function for θ. Propose a transformation of this likelihood function whose maximum is the same and can be computed easily.
An actual experiment is performed, the infrastructure team has obtained the bandwidth totals to failure given by the sequence ξ10 of numbers. Estimate the model-parameter with the maximum likelihood and compute the expectation of the bandwidth total to failure of the dual-router-system.
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