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Important Random Variables

Important Random Variables. EE570: Stochastic Processes Dr. Muqaibel Based on notes of Pillai See also http://www.math.uah.edu/stat http ://mathworld.wolfram.com Check ‘ pdf ’, ‘ cdf ’ commands in Matlab. Continuous-type random variables

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Important Random Variables

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  1. Important Random Variables EE570: Stochastic Processes Dr. Muqaibel Based on notes of Pillai See also http://www.math.uah.edu/stat http://mathworld.wolfram.com Check ‘pdf’, ‘cdf’ commands in Matlab

  2. Continuous-type random variables 1. Normal (Gaussian): X is said to be normal or Gaussian r.v, if This is a bell shaped curve, symmetric around the parameter , and its distribution function is given by where is often tabulated. Since depends on two parameters and , the notation will be used to represent the above CDF Thermal noise : Electronics, Communications Theory

  3. Normal (Gaussian) • is referred to as Standard Normal r.v. • Under very general conditions the limiting distribution of the average of any number of independent, identically distributed random variables is norma.

  4. 2. Uniform: if Quantization Coding Theory 1

  5. Exponential Queuing Theory • If occurrence of events over non-overlapping intervals are independent, such as arrival time of telephone calls or bus arrival times at a bus stop, then the waiting time is exponential. • Memoryless Property of exponential distribution • Memoryless property simplifies many calculations and is mainly the reason for wide applicability of the exponential model. % Dr. Ali Muqaibel close all clear all clc x=-1:0.01:10 y1=pdf('exp',x,1); y2=pdf('exp',x,2); plot(x,y1,x,y2,':'); legend ('prameter=1','parameter=2') xlabel('x') ylabel('f_X(x)') Label ('Exponential Distribution')

  6. 4. Gamma: with (if If an integer then 5. Beta: if where the Beta function is defined as Gamma is a generalization of the exponential distribution with two parameters . If , we get the exponential r.v. Queuing Theory

  7. 6. Chi-Square:  if (Fig. 3.12) Note that is the same as Gamma 7. Rayleigh:  if (Fig. 3.13) 8. Nakagami – m distribution: (3-36) Fig. 3.12 (3-37) Fig. 3.13 In communication systems, the signal amplitude values of a randomly received signal usually can be modeled as a Rayleigh distribution Wireless Communications

  8. 9. Cauchy:  if (Fig. 3.14) 10. Laplace: (Fig. 3.15) 11. Student’s t-distribution with n degrees of freedom Fig. 3.15 Fig. 3.14 Fig. 3.16 Related to Gaussian, Comm. Theory

  9. 12. Fisher’s F-distribution (3-42) Other distributions: Erlang (traffic), Weibull (failure rate), Poreto ( Economics , reliability), Maxwell (Statistical) The exponential model works well for inter arrival times (while the Poisson distribution describes the total number of events in a given period)

  10. Discrete-type random variables 1. Bernoulli: X takes the values (0,1), and 2. Binomial: if (Fig. 3.17) 3. Poisson:  if (Fig. 3.18) (3-43) Fig. 3.18 Fig. 3.17 The total number of favorable outcomes is binomial r.v. The number of occurrence of a rare event in a large number of trials: e.g number of telephone calls at an exchange over a fixed duration

  11. 4. Hypergeometric: 5. Geometric:  if 6. Negative Binomial: ~ if 7. Discrete-Uniform: The number of trials needed to the first success in repeated Bernoulli trials is geometric The number of trials needed to the success in repeated Bernoulli trials is negative binomial

  12. Matlab 'beta' or 'Beta', 'bino' or 'Binomial', 'chi2' or 'Chisquare', 'exp' or 'Exponential', 'ev' or 'Extreme Value', 'f' or 'F', 'gam' or 'Gamma', 'gev' or 'Generalized Extreme Value', 'gp' or 'Generalized Pareto', 'geo' or 'Geometric', 'hyge' or 'Hypergeometric', 'logn' or 'Lognormal', 'nbin' or 'Negative Binomial', 'ncf' or 'Noncentral F', 'nct' or 'Noncentral t', 'ncx2' or 'Noncentral Chi-square', 'norm' or 'Normal', 'poiss' or 'Poisson', 'rayl' or 'Rayleigh', 't' or 'T', 'unif' or 'Uniform', 'unid' or 'Discrete Uniform', 'wbl' or 'Weibull'. Check ‘pdf’, ‘cdf’ commands in Matlab Check rand, randn

  13. Converting Data to PDF % Dr. Ali Muqaibel close all clear all clc n=10000; f=randn(1,n); [y,x]=hist(f,10); y=y/n/(x(2)-x(1)); xm=-1:0.01:10; ym=pdf('norm',xm,0,1); [yr,xr]=ksdensity(f); plot(x,y,xm,ym,xr,yr,':'); legend ('Hist','Model','KSDensity') xlabel('x') • Matlab live demo • Impact of number of points • Difference between histogram and pdf ; Normalization • Fitting

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