Multinomial Probability Function-ECO6416

# Multinomial Probability Function-ECO6416 - Multinomial...

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Multinomial Probability Function A multinomial random variable is an extended binomial. However, the difference is that in a multinomial case, there are more than two possible outcomes. There are a fixed number of independent outcomes, with a given probability for each outcome. The Expected Value (i.e., averages): Expected Value = μ = Σ (X i × P i ), the sum is over all i's. Expected value is another name for the mean and (arithmetic) average. It is an important statistic, because, your customers want to know what to “expect”, from your product/service OR as a purchaser of “raw material” for your product/service you need to know what you are buying, in other word what you expect to get: To read-off the meaning of the above formula, consider computation of the average of the following data 2, 3, 2, 2, 0, 3 The average is Summing up all the numbers and dividing by their counts: (2 + 3 + 2 + 2 + 0 + 3) / 6 This can be group and re-written as: [ 2(3) + 3(2) + 0(1)] / 6 = 2(3/6) + 3(2/6) + 0(1/6) which is the sum of each distinct observation times its probability. Right? Expected value is known also as the First Moment, borrowed from Physics, because it is the point of balance where the data and the probabilities are the distances and the weights, respectively. The Variance is: Var(X) = E[(X- μ ) 2 ] = E[X 2 - 2X μ + μ 2 ]. We simplify this using the above rules. First, because the expectation of a sum equals the sum of expectations: Var(X) = E[X 2 ] - E[2X μ ] + E[ μ 2 ]. Then, because constants may be taken out of an expectation:

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Multinomial Probability Function-ECO6416 - Multinomial...

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