1、Basic Concepts of Probability and Random Variables II7th week / Probability and Statistics (I)Probability Density Function PDF (probability density function) Distribution function for continuous RVs For a continuous RV, probability mass (r = ) is always 0, but probability density is not = r X + / wh
2、en 0= / A PDF is the differentiation of a CDF2Probability Density Function 3Probability Density Function() 4Summary of Probability Distribution FunctionsPMF PDF 0 0 1 1 (?) = = = 1 = 1r = (, = r = = 5Expectation Expectation: a fixed value that represents the value of a random variable Discrete RV: =
3、 = Continuous RV: = = Example: rolling a dice 1 1/6 + 2 1/6+ + 6 1/6 = 3.56Sample Mean Sample Mean: an average of random samples from repeated experiments Example: rolling a dice From 5 experiments, you get 6, 4, 1, 4, and 2 The sample mean is 3.4 The sample mean gets close to the expectation with m
4、ore experiments Law of large numbers7Expectation of Derived Random Variables For function () and RV , = () is a derived random variable from X : the result of rolling a dice = (X -2)2 = 0, 1, 4, 9, 16 , =1/6 for =0, 4, 9, 16, =1/3 for =1 = 5 In general, () = () 8Variance Variance: a fixed value that
5、 represents how much a RV can vary = = = Discrete RV: = () Continuous RV: = () Standard deviation = = 9Properties of Expectation and Variance = : expectation of a constant + = + : linearity Var 0 : non-negative variance Var = 0 : variance of a constant Var+ = 2Var : not affected by offset10SummaryA
6、random variable is an abstract of random values, which is defined by probability distributionCDF: = Pr PMF: = Pr = PDF: = ()/SummaryExpectation: representing the value of a RV = Variance: representing how much a RV varies = 2ReferencesProbability and Stochastic Processes: A Friendly Introduction to Electrical and Computer Engineers (3rd edition), Yates and Goodman, WileyProbability, Statistics, and Random Processes for Electrical Engineering (3rd edition), Leon-Garcia, Pearson International Edition.
