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,
AddisonWesley, 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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 Fall '13
 M.SamiFadali
 Electrical Engineering, Normal Distribution, Probability theory, CDF

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