Random Variables By: 1
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
Probability Mass Function (PMF): Discrete R.V. Probability distribution of X Notation: Properties: 3
Probability Density Function (PDF): Continuous R.V. A continues r.v. is described by a probability density function f X Properties: Interpretation: 4
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
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
Variance: Discrete R.V. Recall: Second Moment: Variance: Properties: 7
Mean and Variance: Continuous R.V. Example: Continuous Uniform r.v. 8
Cumulative Distribution Function (CDF) Discrete r.v. Continuous r.v. Example: 9
Mixed Distributions 10
Gaussian (Normal) PDF 11 Standard Normal: Bell shaped curve: Expectation and variance: General Normal: Expectation and variance: