Discrete Probability Distribution Experiment:Toss 2 Coins.Let X =heads. 4 possible outcomes Probability Distribution X Value Probability 0 114=.25 2/4=.50 2 114=.25 50 2 0 1 2 X Chap 5-6 Tongji University School of Economics Management Statistics for Managers Using Microsoft Excel
Statistics for Managers Using Microsoft Excel Chap 5-6 Tongji University School of Economics & Management Probability Distribution 0 1 2 X X Value Probability 0 1/4 = .25 1 2/4 = .50 2 1/4 = .25 .50 .25 Probability Experiment: Toss 2 Coins. Let X = # heads. T T Discrete Probability Distribution 4 possible outcomes T T H H H H
Practices in Class -Practice:Let the random variable X represents the number of boys in a family,construct the probability distribution for a family of three children 89 P(X)） 0 1/8 1 3/8 13 2 3/8 3 1/8 Chap 5-7 Tongji University School of Economics Management Statistics for Managers Using Microsoft Excel
Statistics for Managers Using Microsoft Excel Chap 5-7 Tongji University School of Economics & Management Practices in Class ▪ Practice: Let the random variable X represents the number of boys in a family, construct the probability distribution for a family of three children X P(X) 0 1/8 1 3/8 2 3/8 3 1/8
Discrete Random Variable Summary Measures Expected Value of a discrete distribution (Weighted Average) N H=EX)=∑XP(X) X P(X) Example:Toss 2 coins, 0 .25 X=#of heads, .50 compute expected value of X: .25 E(X)=(0x.25)+(1×.50)+(2x.25) =1.0 Chap 5-8 Tongji University School of Economics Management Statistics for Managers Using Microsoft Excel
Statistics for Managers Using Microsoft Excel Chap 5-8 Tongji University School of Economics & Management Discrete Random Variable Summary Measures ▪ Expected Value of a discrete distribution (Weighted Average) ▪ Example: Toss 2 coins, X = # of heads, compute expected value of X: E(X) = (0 x .25) + (1 x .50) + (2 x .25) = 1.0 X P(X) 0 .25 1 .50 2 .25 = = = N i 1 i i E(X) X P(X )
Discrete Random Variable Summary Measures (continued) Variance of a discrete random variable W o2=∑X,-EX)PPX,) i=1 Standard Deviation of a discrete random variable N =V2=∑X-EXPX) where: E(X)=Expected value of the discrete random variable X Xi the ith outcome of X P(Xi)=Probability of the ith occurrence of X Chap 5-9 Statistics for Managers Using Microsoft Excel Tongji University School of Economics Management
Statistics for Managers Using Microsoft Excel Chap 5-9 Tongji University School of Economics & Management ▪ Variance of a discrete random variable ▪ Standard Deviation of a discrete random variable where: E(X) = Expected value of the discrete random variable X Xi = the ith outcome of X P(Xi ) = Probability of the ith occurrence of X Discrete Random Variable Summary Measures = = − N i 1 i 2 i 2 σ [X E(X)] P(X ) (continued) = = = − N i 1 i 2 i 2 σ σ [X E(X)] P(X )
Discrete Random Variable Summary Measures (continued) Example:Toss 2 coins,X =heads, compute standard deviation (recall E(X)=1) V∑X-E(X]P(X) 0=V0-1)2(.25)+(1-1)2(.50)+(2-1)2(.25)=V.50=.707 X P(X) 0 .25 Possible number of heads 1 .50 =0,1,0r2 2 25 Chap 5-10 Tongji University School of Economics Management Statistics for Managers Using Microsoft Excel
Statistics for Managers Using Microsoft Excel Chap 5-10 Tongji University School of Economics & Management ▪ Example: Toss 2 coins, X = # heads, compute standard deviation (recall E(X) = 1) Discrete Random Variable Summary Measures σ [X E(X)] P(X )i 2 i = − σ (0 1) (.25) (1 1) (.50) (2 1) (.25) .50 .707 2 2 2 = − + − + − = = (continued) X P(X) 0 .25 1 .50 2 .25 Possible number of heads = 0, 1, or 2