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MIME 4100 HWK 4 2/10/2014 Due 2/19/2014 1. Read Sections C.2.4. and C.3 2. The joint PMF of random variables X and Y is shown below. Specifically, each cell below shows the probability of variables X and Y assuming nine pairs of values. For example the probability of X=2 and Y=3 is 0.08. X\Y 1 2 3 a) b) c) d) 1 0.16 0.16 0.08 2 0.16 0.16 0.08 3 0.08 0.08 0.04 Value of PMF for X=2 and Y=3 Find the marginal probability density functions of X and Y. Determine if X and Y are independent. Compute the covariance and correlation coefficient of the two random variables. Compute the probability that X is greater or equal to Y. 3. In problem 1, if you are told that Y=1, what is the conditional probability of X? 4. Fill out the blanks in the statements below: a. An estimator is unbiased if ____________________ b. An estimator is consistent if ____________________ c. The sample variance is given by the following equation ______________ d. The sample proportion of an event B is given by the following equation ______________________ 5. The observations shown below have been recorded in an experiment. Calculate the sample mean, variance and the sample proportion of values greater that 1.5. Hint: To reduce the amount of calculations copy and paste the numbers below to a MS-Excel spreadsheet and use the functions in Excel to do the computations. 0.90993 0.616695 1.073277 1.382942 1.359505 1.51994 0.344924 0.929746 1.328507 0.67399 0.792939 0.49287 0.445927 0.706711 0.767948 0.364621 0.829623 0.878786 1.040456 0.890352 0.901903 0.888928 1.402792 0.974415 0.944153 0.846038 1.591664 1.259702 1.712696 0.803528 6. Test the hypothesis that the data in problem 5 have been drawn from a normal (Gaussian) PDF with mean value one and standard deviation 0.4. Divide the data into the following four intervals [0,0.5], [0.5, 1], [1,1.5],[1.5,2].