1. Important Random Variables EE570: Stochastic Processes Dr. Muqaiebl Based on notes of Pillai See also http://www.math.uah.edu/stat http://mathworld.wolfram.com

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 (3-29). (3-29) (3-30) Fig. 3.7 Thermal noise : Electronics, Communications Theory

3. 2. Uniform:  if (Fig. 3.8) (3.31) Fig. 3.8 3. Exponential:  if (Fig. 3.9) (3-32) Fig. 3.9 Queuing Theory Coding Theory

4. 4. Gamma:  if (Fig. 3.10) If an integer 5. Beta:  if (Fig. 3.11) where the Beta function is defined as (3-33) (3-34) (3-35) Fig. 3.11 Fig. 3.10 Queuing Theory

5. 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) (3-37) Fig. 3.12 Fig. 3.13 (3-38)

6. 9. Cauchy:  if (Fig. 3.14) 10. Laplace: (Fig. 3.15) 11. Student’s t-distribution with n degrees of freedom (Fig 3.16) Fig. 3.14 (3-41) (3-40) (3- 39) Fig. 3.15 Fig. 3.16 Related to Gaussian, Comm. Theory

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

8. 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) (3-44) (3-45) Fig. 3.17 Fig. 3.18

9. 4. Hypergeometric: 5. Geometric:  if 6. Negative Binomial: ~ if 7. Discrete-Uniform: We conclude this lecture with a general distribution due (3-49) (3-48) (3-47) (3-46)