Suppose I have a multinomial distribution with 10 outcomes and each has prob of 0.1 (i.e. weight of 10%). After each random draw, I can calculate the weight of the sample w1, w2... w10, and I want to calculate the expected deviation from the mean: $E((w_1-0.1)^2 + (w_2-0.1)^2... + (w_10-0.1)^2)$. Somehow I feel like I can reduce it to summing the variance of each individual outcome. But what happen to the covariance term? https://en.wikipedia.org/wiki/Multinomial_distribution I am confused and feel I am missing something.
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Deviation from the mean of what? A multinomial outputs an array of frequencies, e.g. across 10 colors. – Aaron Goldsmith Dec 18 '22 at 04:37
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@AaronGoldsmith the total deviation from the expected weights of each color, which is 10%: $((1−0.1)^2+(2−0.1)^2...+(10−0.1)^2)$ – ian_chan Dec 18 '22 at 05:20
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Since $w_1,\dots, w_{10}$ are identical, use linearity of expectation to get in terms of only $w_1$. Then, it becomes a binomial problem. – Aaron Goldsmith Dec 19 '22 at 16:45