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{"ops":[{"insert":"PROBLEM: \nSuppose a sample x"},{"attributes":{"script":"sub"},"insert":"1"},{"insert":", ..., x"},{"attributes":{"script":"sub"},"insert":"n"},{"insert":" is modelled by a "},{"attributes":{"bold":true},"insert":"Poisson distribution"},{"insert":" with parameter denoted \u03bb, so that\n"},{"attributes":{"bold":true},"insert":"\u00a0"},{"insert":"\n                    \n"},{"attributes":{"bold":true},"insert":"\u00a0"},{"insert":"\n(a)\u00a0\u00a0\u00a0Restate the likelihood function of a Poisson distribution\n\u00a0\n(b)\u00a0\u00a0Prove that the natural log likelihood function of the Poisson distribution i.e. show that you arrive at this log function.\n\u00a0\n  \n(c)\u00a0\u00a0\u00a0Expand, rearrange, and take first derivate and show that you arrive at the function below\n  \n(d)\u00a0\u00a0Find the maximum likelihood function (Hint: make equation in (c) equal to 0.\n\u00a0\n(e)\u00a0\u00a0\u00a0Let us assume a dataset with n= 647 and the observed frequencies of domestic accidents are as follows\n\u00a0\n Number of accidents\t\t\t\tFrequency\n\t\t0\t\t\t\t\t\t\t\t\t\t\t447\n        1\t\t\t\t\t\t\t\t\t\t\t132\n        2\t\t\t\t\t\t\t\t\t\t\t42\n        3\t\t\t\t\t\t\t\t\t\t\t21\n        4\t\t\t\t\t\t\t\t\t\t\t3\n        5\t\t\t\t\t\t\t\t\t\t\t2\n\n\n \nUsing the Poisson distribution, calculate the MLE for \u03bb i.e. prove that MLE for \u03bb is 0.465.\n"}]}

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