1. Random Variables By: www.entcengg.comwww.entcengg.com 1

2. Random Variables  An assignment of a value (number) to every possible outcome.  Mathematically: A function from the sample space Ω to the real numbers. − discrete or continuous values.  Can have several random variables defined on the same sample space.  Notation: − random variable X − Numerical value x 2

3. Probability Mass Function (PMF): Discrete R.V.  Probability distribution of X  Notation:  Properties: 3

4. Probability Density Function (PDF): Continuous R.V.  A continues r.v. is described by a probability density function f X  Properties:  Interpretation: 4

5. Expectation: Discrete R.V.  Definition:  Interpretation: − Center of gravity of PMF − Average in large number of repetitions of the experiment  Example: Uniform on 0,1, 2,…, n. Find E(X) 5

6. Properties of Expectation  Let X be the r.v. and let Y = g(X)  Caution: In general,  Properties: If α and β are constants, then: 6

7. Variance: Discrete R.V.  Recall:  Second Moment:  Variance:  Properties: 7 www.entcengg.com

8. Mean and Variance: Continuous R.V.  Example: Continuous Uniform r.v. 8

9. Cumulative Distribution Function (CDF)  Discrete r.v.  Continuous r.v.  Example: 9

10. Mixed Distributions 10

11. Gaussian (Normal) PDF 11  Standard Normal:  Bell shaped curve:  Expectation and variance:  General Normal:  Expectation and variance: www.entcengg.com