 # Probability Questions

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Problem 1 Let x be one observation from a N(0,0²) population. Is |X| a sufficient statistic? Problem 2 Let X1. X= be independent random variables with densities = 0 Prove that T = is a sufficient statistic for 0. Problem 3 One observation is taken on a discrete random variable x with pmf f(x|0), where OE {1,2,3}. Find the MLE of 0. x f(x|1) f(x|2) f(x)3) 1/3 1/4 1 1/3 1/4 2 1/4 1/4 3 1/6 1/4 1/2 4 1/6 1/4 Problem 4 Let X1 X. be i.i.d with Xi n Poisson(A), r > 0. Compute the ML estimator for A. Problem 5 Assume that a gas station needs at least a > 0 minutes for an oil change, where a is some constant. The actual time that an oil change requires varies from case to case. Suppose you want to model the required time for an oil change by a random variable x = Z+a, where Z is a random variable that satisfies Z 0. Suppose that Z N Exponential (1), i.e., the density of Z is given by (a) What is the density of X? 1 (b) Suppose that you have 10 independent observations of times required for an oil change: 1.2,3.1,3.6,4.5,5.1,7.6,4.4,3.5,3.8,4 Based on these data, compute the ML estimate for a. Problem 6 During the preparation for an exam Bob and Anna communicate via e-mail to clarify arising questions. Due to the time points that Anna receives e-mails from Bob she is afraid that he works more than she does. The time points (on a 24 hour scale) of the last ten e-mails are 10.55, 14.9, 11.2, 18.85, 9.75, 11.5, 16.1, 14.4,9.2, 12.95; Based on these data Anna wants to estimate the daily working time of Bob. For this pur- pose she assumes that Bob starts studying each day at a fixed time a and works until a fixed time b > a. Furthermore, she assumes that the time points when Bob sends e-mails are inde- pendent and uniformly distributed on the interval [a, bj. Can you help Anna and estimate for her the parameters a and b with (a) ML estimation; (b) The method of moments, i.e., find those parameters a, b for which the mean and the vari- ance of a Ua, b] distribution coincide with the sample average and the sample variance, respectively. What do you think about these estimation procedures? 2

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