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2 years ago in Discrete Mathematics , Statistics By Pavithra sp

How to compute expectation and variance for constrained integer sets?

I’m analyzing a dataset where each entity has a non-negative integer count, and these counts must sum to a fixed total. Calculating the naive sample mean and variance ignores this constraint, biasing my model. I need to know the correct way to compute the expectation and variance that properly accounts for this inherent dependency in the data structure.

SWaP_Constraints Variance

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By Natasha Answered 1 year ago

This is a classic issue in multivariate analysis and combinatorics. You cannot treat constrained k_i as independent samples. The key is to define the correct sample space. I would recommend modeling your data as a draw from a multivariate distribution, like the multinomial if the total sum N is fixed. The expectation for each category is then N * p_i, and the variance is N * p_i * (1-p_i), with covariance between categories. If p_i is unknown, use the observed proportion. For just non-negativity without a fixed sum, treat it as a Poisson-binomial scenario. The constraint fundamentally changes the covariance structure.

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How to compute expectation and variance for constrained integer sets?

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