Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Chapter 5 Discrete Probability Distribution 5.1 Random Variables Discrete discrete values only Continuous any values on a interval Random variables is a function which assign a real number to each element of a sample space. One sample space may be corresponding to more than one random variables. e.g. Consider throwing 3 coins random variable “no. of T shown” 0 1 2 3 random variable sample space “no. of heads shown” HHH 3 HHT HTH 2 THH HTT THT 1 TTH TTT 0 5.2 Probability Distributions It can be shown by 1. a table 2. a line graph(bar chart) 3. a probability histogram - forget it 4. a mathematical formula(a function) - pdf Probability Function(pdf) A pdf must satisfy 1. 0 f(x) for all x A f (x) 2. 1 P(X= x) = f(x) A is the set of all possible values xA 5.3 Expected Value (Expectation) e.g. Consider throwing a die a frequency distribution of throwing 120 times x f 1 20 2 19 3 20 4 21 5 21 6 19 = a probability distribution x prob. 1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/6 1 fi xi N expected value = Pixi f = ( i ) xi N = E(X) = f(x)x E(aX+b) = a E(X) + b IIT_web_material 5.4 Variance & Standard Deviation recall the formula for frequency distribution 1 2 = fi (xi - )2 N f = ( i ) (xi - )2 N variance of a discrete random variable X Var(X) = 2 = f(x) (x - )2 = E [ (X - )2 ] Var (a) =0 Var(aX) = a2 Var(x) aX = a X Var(aX + b) = a2 Var(X) IIT_web_material other formula 1 2 = fi xi - 2 N Var(X) = 2 = f(x)x2 - 2 = E(X2) - 2