Proof of Normality of Distribution of a Subset of Random Variables Based on the Transformation of Block Matrices
https://doi.org/10.1134/S2304487X21010107
Abstract
The multidimensional normal distribution of random variables is one of the main distributions in solving a large number of statistical problems. It is well known that the joint marginal distribution of a subset of random variables is also normal. In the literature, this fact is proved by finding the characteristic function for a given set of random variables and then finding the characteristic function for the subset of random variables and comparing the characteristic functions. Using this approach, it is relatively easy to prove that the subset of random variables belongs to the normal distribution. However, the density function of the multivariate normal distribution is not found explicitly, the vector of mathematical expectations and the covariance matrix of the distribution are not calculated. It would be interesting to obtain a rigorous proof that if the original set of random variables has a multidimensional normal distribution, then the subset of random variables of this set also has a multidimensional normal distribution with certain parameters. This work provides a rigorous derivation of the multivariate normal distribution density function for the subset of random variables. The vector of mathematical expectations and the covariance distribution matrix are calculated. The inference is based on block matrix operations. The presented formulas for calculating the inverse matrix, when the original matrix is presented in the form of blocks, are of independent interest.
About the Author
K. Y. Kudryavtsev
Institute of Cyber Intelligence Systems, National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
115409
Moscow
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Review
For citations:
Kudryavtsev K.Y. Proof of Normality of Distribution of a Subset of Random Variables Based on the Transformation of Block Matrices. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2021;10(1):70-76. (In Russ.) https://doi.org/10.1134/S2304487X21010107
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