APPENDIX B Multivariate Statistics Joint and Multivariate Distributions Independence and Correlation Bivariate Gaussian Distribution Central Limit Theorem Huseyin Bilgekul EEE 461 Communication Systems II Department of Electrical and Electronic Engineering Eastern Mediterranean University

Joint or Multivariate CDFs N-dimensional CDF and PDF are: Joint CDF and PDF of two RVs

Marginal PDFs Can obtain individual or MARGINAL PDFs by integrating across the entire range of the other axis: Using the Bayes Rule Note that P[X = x] = 0, but {X = x} has occurred (X must have some value) and fX(x)  0. Independent RVs mean that one variable does not depend on the other. When this property holds it makes analysis much easier.

Independence of RV’s INDEPENDENT random variables: For N independent RV’s Two random variables are ORTHOGONAL if:

Correlation or Joint Mean of RV CORRELATION or Joint Mean is: Two RV’s are UNCORRELATED if NOTE: If RV’s are independent, then they are uncorrelated, not vice versa Two RV’s are ORTHOGONAL if

Covariance and Correlation Coefficient COVARIANCE is defined as: Covariance is zero if two RV’s are INDEPENDENT but the inverse is not generally true. Correlation Coefficient If x=y, then variables are correlated, x=-y, rho = -1. If x and y are independent then rho=0 We will use correlation to design detectors for coherent receivers.

Correlation Correlation is used to determine how closely one input resembles another Used in matched filter type demodulators to decide which waveform was sent ρ=1 ρ =0 ρ = -1

Correlation-Type Demodulators Decomposes the received signal and noise into N-dimensional vectors Received signal: R(t) = si(t) + n(t) Multiply the signal by a “basis function” that spans the space x To detector

Correlation Receivers Received signal after demodulation is: Branch with the highest match is the detected output Derivation requires some understanding of random processes so we will do later

Bivariate Normal Distribution The bivariate Gaussian PDF is: If bivariate Gaussian RV’s are uncorrelated then they are independent. If r=0, f(x,y) = f(x) f(y)

Bivariate Normal ρ =0 ρ = 0.5

Central Limit Theorem Sums of independently distributed RVs tend to become a Gaussian Distribution Holds for most distributions as long as they are unimodal (one hump) and tend to zero at +/- infinity One of the main reasons that the Normal distribution is used

Central Limit Theorem Under certain conditions, sum of a large number of RVs tends to be a Gaussian RV If z = x+y, then fz(z) = fx(x)* fy(y) Why? The theorem In practice, n  6 is big enough!

PDF for Sum of RVs is Convolution Same logic holds for sums of more RVs If z = x+y, then fx(z) = fx(x)* fy(y) With each additional convolution the result approaches the normal distribution x1 fx1 x1+ x2 f (x1+ x2) fx2 x2

Example Rayleigh Distribution Given two independent, identically distributed (IID) Gaussian RVs, x and y: Find the PDFs of the amplitude and phase of these variables (polar coordinates):