(X2-E(X2))2/E(X2) + … random sample of outcomes of type 2, … , of the normal curve and Student's t-curve: n2 outcomes of type 2, … , Since (0.1) is well known, the proof is omitted. the probability histogram of the chi-squared … + pk = 100%. Draw 10,000 samples of size ' + rolls.toString() + ' and see how often ' + where oi is the number of elements of the random sample that This leads to the summary statistic, Let oi denote the number times an outcome in Let us look at the p1n1 × and repeat the experiment for different sample sizes. category. … , 6, ways to allocate the n2 outcomes of type 2 among 0 and 1, the a quantile of the chi-square curve with There are two proofs of the multinomial theorem, an algebraic proof by induction and a combinatorial proof by counting. categories. (1/6)0 × (1/6)0×(1/6)0 a is the desired significance level. xk-1,1-a, the observed numbers of outcomes in each category and the expected number of outcomes in each chi-square statistic. Specifically, (o1 - 5×0.1)2/(5 × 0.1) {X1, X2, …, If the number of categories or their probabilities vary from trial to trial, As the degrees of freedom increases, the peak moves to larger and larger values, Then {X1, X1, X4 = 0, X5 = 0, X6 = 0). data arises from a multinomial distribution with k categories and of categorical data match the frequencies that would be expected under the null and your ability to calculate the chi-squared statistic. (X1-np1)2/(np1) + of counting, the total number of ways is therefore. in category k, This is called the multinomial distribution with parameters n and Xk = nk ) the n1 outcomes of type 1 among the n trials. n×pi>10 for every category A multinomial trials process is a sequence of independent, identically distributed random variables X=(X1,X2,…) each taking k possible values. '(' + outcomes[5].toString() + ' - ' + expStr + ')2/' + expStr + var expect = rolls/6; If we think of the chi-squared curve with d degrees of This is called the chi-square test for goodness of fit. histogram of the chi-squared statistic. (For example, Chebychev's inequality bounds the chance 'the following sense: as d grows, the area under the ' + n-n1-n2 trials, etc. In an independent sequence of n trials,