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Can be written in terms of variances and correlation

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Unformatted text preview: amp; variance to vector case. • Can be written in terms of variances and correlation coefficients. • Diagonal for uncorrelated (& independent) variables. 51 52 Multivariate Normal / Proof: • Generalization of normal distribution to n linearly independent random variables. • If are mutually uncorrelated they are also independent. Quadratic form / 53 Bivariate Gaussian 54 Properties of Multivariate Normal • Density N completely defined by and . • If joint pdf is normal: – Uncorrelated Independent. – All marginal and conditional pdfs are normal. • Linear transformation of normal vector gives a normal vector (next presentation). 55 56 References Conclusion • Probabilistic description of random variables. • Moments, characteristic function, moment generating function. • Correlation and covariance. • Correlated, independent, orthogonal. • Normal (Gaussian) random variable. 57 • Brown & Hwang, Introduction to Random Signals and Applied Kalman Filtering, Wiley, NY, 2012. • Stark & Woods, Probability and Random Processes, Prentice Hall, Upper Saddle River, NJ, 2002. • R. M. Gray & L. D. Davisson, Random Processes: A mathematical Approach for Engineers, Prentice Hall, Englewood Cliffs, NJ, 1986. • M. H. De Groot, M. J. Schervish, Probability & Statistics, Addison-Wesley, Boston, 2002. • S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory, Prentice Hall, 1998. • A. Papoulis and S.U. Pillai, Probability, Random Variables, and Stochastic Processes, 4th Ed., McGraw Hill, Boston, 58 MA, 2002....
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