1. 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

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

3. 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.

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

5. 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

6. 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.

7. 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

8. 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

9. 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

10. 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)

11. Bivariate Normal
ρ =0 ρ = 0.5

12. 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

13. 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!

14. 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

15. 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):