1. Useful Discrete Random Variable
The University of Texas at Dallas
Useful Discrete Random Variable
Duncan MacFarlane
The University of Texas at Dallas

2. Distributions and their uses
Bernoulli – Number of successes on one trial
Geometric – Number of trials until first success
Binomial – Number of successes on n trials
Pascal – Number of trials until the k-th success
Poisson – Arrival statistics

3. Bernoulli and Geometric Distributions
Number of successes on one trial
P(x) = (1-p) for x=0, or
P(x) = p for x=1
Geometric
Number of trials until first success
P(x) = p(1-p)(n-1)

4. Geometric PMF

5. Geometric CDF

6. Binomial Distribution
Binomial – Number of Successes on n trials

7. Binomial PMF, n=10

8. Binomial CDF n=10

9. Pascal Distribution
Pascal – Number of trials until the k-th success
For n=k, k+1, k+2 ….

10. Arrival Statistics (busses, customers, etc)
Poisson Distribution
Arrival Statistics (busses, customers, etc)
A: is the average arrival rate times the observation time
A=T

11. Poisson PMF

12. Poisson CDF

13. Distribution E(X) Var(X)
Summary
Distribution
E(X)
Var(X)
Bernoulli p
p(1-p)
Geometric
1/p
(1-p)/p2
Binomial np
np(1-p)
Pascal
k/p
p(1-p)/p2
Poisson A