7:12 PM P P . D .Ill 11 3011 84 25 111 1618253302216_Sta... . . . 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 an exponential distribution whose density is given by: with a single parameter 0. Consider the bandwidth total to failure ?' 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 S, and $2. 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 Ti, 72. ..., Tm, calculate the likelihood function for e. 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 following bandwidth total to failures: 9.2, 5.6, 18.4, 12.1, 10.7 Grate Winds Estimate the model-parameter with the maximum likelihood and compute the expectation of the bandwidth total to failure of the dual-router-system. Workbook Assignment 4: Hypothesis Test: 15% Over a long period of time, the production of 1000 high-quality hammers in a factory seems to have reached a weight with an average of 971g and standard deviation of 15.2 g. Propose a model for the weight of the hammers including a probability distribution for the weight. What are the assumptions for this model to hold? What parameters does this model have? A new production system is configured, and one wants to evaluate if the new system makes more constant weights. For this a random sample of newly produced hammers is evaluated yielding the following weights: 987, 966, 955,977, 981,967, 975, 980, 953, 972 What hypothesis can you formulate and what test and decision rule can you make to estimate if the new system produces a more constant weight? Express these assertions as logical statements involving critical values. What error probabilities can you suggest and why? Calculate the p-value. Perform the test and express conclusions. Workbook Assignment 5: Sufficient Statistics: 15% (following Larsen & Marx, exercise 5.6.5) Let X1, X2, ". Xm be the random sample of a positive random variable X having the density function: fx (x: 0) = 51 06xle With one parameter 8 6 R. Find an estimator for @ that is sufficient. Workbook Assignment 6: Bayesian Estimates: 15% (following Hogg, Mckean & Craig, exercise 11.2.2) Let X1, X2, ". Xjo be a random sample from a gamma distribution with a = 3 and # = 1/0. Suppose we believe that follows a gamma-distribution with a = 3 and 8 = 2: Activate Winda Workbook Assignment 6: Bayesian Estimates: 15% following Hogg, Mckean & Craig, exercise 11.2.2) Let X1, Xzr w Xjo be a random sample from a gamma distribution with a = 3 and B = 1/0. Suppose we believe that follows a gamma-distribution with a = 3 and B = 2: a) Find the posterior distribution of 0. b) If the observed x = 18.2, what is the Bayes point estimate associated with the square-error loss function? c) What is the Bayes point estimate using the mode of the posterior distribution? O
