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MGSC 372 Lecture Notes - Lecture 2: Maximum Likelihood Estimation, Log-Normal Distribution, Estimation Theory

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10 May 2017

School

McGill University

Department

Management Science

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Document Summary

A discrete example mle for binomial distribution (cid:887) Likelihood function for three values of :{(cid:2869),(cid:2870),(cid:2871)} (cid:4666)(cid:4667)=((cid:2869))(cid:3117)(cid:4666)(cid:883) (cid:4667) (cid:3117) ((cid:2870))(cid:3118)(cid:4666)(cid:883) (cid:4667) (cid:3118) ((cid:2871))(cid:3119)(cid:4666)(cid:883) (cid:4667)(cid:4666) (cid:3119)(cid:4667) (cid:4666)(cid:4667)=((cid:2869))((cid:2870))((cid:2871))(cid:4666)(cid:3117)+(cid:3118)+(cid:3119)(cid:4667) (cid:4666)(cid:883) (cid:4667)(cid:4666) (cid:3117)(cid:4667)+(cid:4666) (cid:3118)(cid:4667)+(cid:4666) (cid:3119)(cid:4667) (cid:4666)(cid:4667)=((cid:2869))((cid:2870))((cid:2871))(cid:3117)+(cid:3118)+(cid:3119) (cid:4666)(cid:883) (cid:4667)(cid:4666)(cid:2871) [(cid:3117)+(cid:3118)+(cid:3119)](cid:4667) ln(cid:4666)(cid:4667)=ln((cid:2869))((cid:2870))((cid:2871))+ln(cid:3117)+(cid:3118)+(cid:3119)+ln(cid:4666)(cid:883) (cid:4667)(cid:4666)(cid:2871) [(cid:3117)+(cid:3118)+(cid:3119)](cid:4667) ln(cid:4666)(cid:4667)=ln((cid:2869))((cid:2870))((cid:2871))+(cid:4666)(cid:2869)+(cid:2870)+(cid:2871)(cid:4667)ln+(cid:4666)(cid:885) [(cid:2869)+(cid:2870)+(cid:2871)](cid:4667)ln(cid:4666)(cid:883) (cid:4667) Generalizing to the case of observations we get: A coin is tossed 10 times and the number of heads is recorded. The number of heads obtained in each of the 8 repetitions of the experiment is as follows: The probability of heads is close to 50% so the coin appears to be fair. An assembly line produces a fixed, but unknown, percentage of defective items. 100 items are selected and the number (%) of defective items items in each sample is recorded as follows: (cid:883)(cid:882) (cid:890) The mle for the percentage of defective items produced by the assembly line is given by: We conclude that =(cid:882). (cid:882)(cid:886)(cid:888) so approximately 4. 6% of items produced.

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MGSC 372 Lecture Notes - Lecture 1: Maximum Likelihood Estimation, Random Variable, Probability Distribution

Related Questions

1.Create a list that contains 10 successive integers starting with the value in the variable start. For example, if start is 7, the resulting list should be [7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. Store the list in a variable named my_list. Assume that the value of start has already been set.

2.Create a list that is made up of the first half of the elements of a list called original_list. Store the new list in a variable called half_list. Assume that the original list has already been initialized and contains an even number of elements.

3. Write a predicate function called same_ends that accepts a list of integers and an integer (n) as arguments. Return True if the first n numbers in the list are the same as the last n numbers in the list, and False otherwise. For example, if the list was [1, 2, 3, 4, 99, 1, 2, 3, 4], the function would return True if n is 4 and False if n is 3. Assume that n is between 1 and the length of the list, inclusive.

4.Write a function called highest_values that accepts as arguments a list of numeric values and a single integer (n). The function should return a list made up of the largest n values from the argument list in order from lowest to highest. For example, if the argument list is [46, 82, 25, 61, 70, 33, 54] and n is 3, the return value should be [61, 70, 82]. Assume that the argument list contains at least n elements.

5.Assume that a list of integers stored in a variable named original contains exactly two elements that are zero. Write code that creates a separate list that contains the elements from original that are between the two zeros. Do not include the zeros in the new list. Store the list in a variable named between_zeros.

6. Write a function called price_range that accepts a list of floating-point numbers representing prices as an argument and returns the the difference between the highest price and the lowest price. Do NOT use the built-in functions max or min in your solution. Assume that all prices are positive values and that none of them exceed 10,000.00.

7. Write a function called count_vowels that accepts a string argument that represents a word and returns the number of vowels that are in the word. The vowels are A, E, I, O, and U (ignore the 'sometimes Y' rule). Count both uppercase and lowercase vowels.

Assume that the variable table holds a two-dimensional list of integers. Write code that computes the sum of all elements in the table, assigning the result to the variable table_sum.

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