Review of Basic Probability and Statistics

1. Review of Basic Probability and Statistics ISE525: Spring 10

2. Random Variables and Their Properties • Experiment : a process whose outcome is not known with certainty. • Set of all possible outcomes of an experiment is the sample space. • Outcomes are sample points in the sample space. • The distribution function (or the cumulative distribution function, F(x), of the random variable X is defined for each real number as follows:

3. Properties of distribution functions 1) 2) F(x) is nondecreasing. 3)

4. Discrete Random Variables • A random variable, X, is said to be discrete if it can take on at most a countable number of values: • The probability that X takes on the value xi is given by • Also: • p(x) is the probability mass function.

5. Discrete variables continued: • The distribution function F(x) for the discrete random variable X is given by:

6. Moments

7. Moments of a Probability Distribution • The variance is defined as the average value of the quantity : (distance from mean)2 • The standard deviation, σ =

8. For discrete Random Variables

9. Continuous random variables • A random variable X is said to be continuous if there exists a non-negative function, f(x), such that for any set of real numbers B, • Unlike a mass function, for the continuous random variable, f(x) is not the probability that the random number equals x.

10. Multiple random variables • IF X and Y are discrete random variables, then the joint probability mass function is: • P(x,y) = P(X=x, Y=y) • X and Y are independent if:

11. Multiple random variables • For continuous random variables, the joint pdf is • For independence:

12. Properties of means • This holds even if the variables are dependent!

13. Properties of variance • This does not hold if the variables are correlated.

14. Common Discrete Distributions • Bernoulli: Coin toss • Binomial: Sum of Bernoulli trials

15. Poisson Distribution:

16. Common Continuous Distributions • Uniform

17. Exponential Distribution • Probability distribution function (pdf) and the Cumulative distribution functions (cdf) are: • Mean and Standard Deviation are:

18. Common Continuous Distributions • Normal Distribution:

19. Other Distributions • Erlang distribution:

20. Gamma Distribution

21. Estimation • Means, variances and correlations: • Simulation data are almost always correlated (according to Law and Kelton) !

22. Hypothesis tests for means

23. Strong law of large numbers