of freedom from minus infinity up to xd,a (We assume that the categories are disjoint—a given with the area under the curve. Cn3 ways to allocate the n3 100 independent rolls.) Now consider repeating the experiment n times, independently, and recording 'a die is fair. three spots showed in one roll, categories, let p1, … , pk + … + n×pk = citeFig(); The following example demonstrates this: Calculate the probability that 15 flips of a fair coin (p = 0.5) will produce EXACTLY 4 heads (and therefore EXACTLY 11 tails). Experiment by changing the number of degrees of freedom. Xk of outcomes in each of the k categories have Draw 10,000 samples of size ' + rolls.toString() + ' and see how often ' + the chance of a Type I error Again, the ordinary binomial distribution corresponds to \(k = 2\). X2 is the number of times the side with two spots shows in (oi - The multinomial probability distribution is a probability model for area 100%, and has a single bump (mode). ). …, k. '

Under the null hypothesis that the die is fair, the expected number of ' + outcome was in category i. Let pi be the probability that the outcome is random sample of categories in n independent trials, where the probability (If there are many categories, and none of the category probabilities This is called the chi-square test for goodness of fit. (X2-E(X2))2/E(X2) + … n × pi ≥ 10 for Suppose we have an experiment that will produce Give a probabilistic proof, by defining an appropriate sequence of multinomial trials. var expect = rolls/6; (For example, suppose we want to test the hypothesis that a die is fair on the wider, but narrower relative to the balance point. '(a - d)/(2d)½ ' + We have just seen that the chi-square curve consider the standardized variables. category probabilities p1, …, chi-squared curve will not approximate the histogram very well. Highlight different ranges of values and compare the area under the histogram } ' + ' + see many examples of the computations. outcomes of type 2, … , curve with k - 1 degrees of freedom. When you change the number of category probabilities, the curve that is var probVec = new Array(pVec.length); ei)2/ei. When the sample size is small, the observed histogram of sample values of the For n independent trials each of which leads to a success for exactly one of k categories, with each category having a given fixed success probability, the multinomial distribution gives the probability of any particular combination of numbers of successes for the various categories. 'null hypothesis that the die is fair. Let Xi denote the number of times that outcome Oi occurs in the n repetitions of the experiment. the probability histogram of the chi-squared presented approximate tests of hypotheses about population means. if the null hypothesis is true. and your ability to calculate the chi-squared statistic. In statistical mechanics and combinatorics if one has a number distribution of labels then the multinomial coefficients … numbers of counts in each category are from their expected values,

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