# Bài giảng Probability & Statistics - Lecture 6: Continuous probability - Bùi Dương Hải

Example Example 6.2. Return (\$mil) of project A is normality with mean of 8 and variance of 9. Calculate the probability: (a) Return of A higher than 10 (b) Loss money (c) Return of A between 5 and 12  Return of project B is normality with mean of 10 and variance of 25. A and B are independent. Calculate the probability that: (c) Both gain positive return (d) Total return of A and B greater than 20

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Lecture 6. CONTINUOUS PROBABILITY  Continuous Random Variable  Density Function  Parameter  Uniform Distribution  Normal Distribution  Cutoff point  [1] Chapter 6. pp. 255 - 294 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 1 6.1. Continuous Random Variable  Continuous Random Variable: uncountable values  Available value is one interval: = (, )  Maybe: = −∞; = +∞  Probability that one point: = = 0  Consider Probability at one interval: ( < < ) PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 2 6.2. Density Function  Discrete  Continuous  ∑ = 1 ∫ = 1 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 3 X Prob. X (, ) Density () f(x)p Density Function  ≥ 0  ∫ = 1  < < = ∫  Cutoff point level denoted by : > = PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 4 f(x) a b 6.3. Parameter  Expected Value: = = ∫  Variance: = ∫ − = ∫ −  Standard Deviation = ()  Cutoff point level , denoted by : > = PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 5 Example Example 6.1. Waiting time (hour), with density function  = 2 ∈ [0,1] 0 ∉ [0,1] (a) Prob. of waiting more than a half of hour? (b) Prob. of waiting from 20 to 40 minutes? (c) The average and variance of waiting time? (d) Cutoff point level 10%? PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 6 Example  (a) > 0.5 = ∫ 2. .  (b) < < = ∫ 2. / /  (c) = ∫ .2. = ∫ .2. − PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 7 f(x) 0.5 f(x) 1/3 2/3 6.4. Uniform Distribution  ~(, ) if  = ∈ [, ] 0 ∉ [, ]  = ; =  < < = Ex. Temperature is Uniform Distribution in the interval of (20, 30)oC. What is the probability that temperature is between 23 and 28 degree? PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 8 a c d b 6.5. Normal Distribution PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 9 0 0.05 0.1 0.15 0.2 0.25 1 2 3 4 5 6 7 8 9 0 0.05 0.1 0.15 0.2 0.25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 0 0.005 0.01 0.015 0.02 0.025 1 6 111621263136414651566166717681869196 0 0.005 0.01 0.015 0.02 0.025 0 20 40 60 80 100  , = 0.5 : = 10; 20; 100 Normality Normal Distribution  Density Function: =  Denoted: ~(, )  =  =  = PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 10 f(x) μ μ’ 1 σ 2π Normal Distribution  Carl Friedrich Gauss (1777-1855) in 1809  ~ 3,1  ~ 6,1  ~(8,0.5 ) PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 11 0 0.1 0.2 0.3 0.4 0.5 0.6 0 1 2 3 4 5 6 7 8 9 10 Standardized Normal Variable  ~ ,  =  ~(0,1)  Table 1  < 1 = 0.8413  < 1.25 =  > 2 =  −1 < < 1.3 = PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 12 0 0.1 0.2 0.3 0.4 0.5 -4 -3 -2 -1 0 1 2 3 4 -4 -3 .5 -3 -2 .5 -2 -1 .5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5 3 3 .5 4 Probability formula  ~ , < = − < − = < − Ex. ~ 100,16  < 104 =  > 92 =  94 < < 102 =  Probability that X differ from the mean not more than standard deviation = PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 13 Example Example 6.2. Return (\$mil) of project A is normality with mean of 8 and variance of 9. Calculate the probability: (a) Return of A higher than 10 (b) Loss money (c) Return of A between 5 and 12  Return of project B is normality with mean of 10 and variance of 25. A and B are independent. Calculate the probability that: (c) Both gain positive return (d) Total return of A and B greater than 20 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 14 3-sigma Rule  − < < + = 68.26%  − 2 < < + 2 = 95.44%  − 3 < < + 3 = 99.75% PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 15 Cutoff point  Cutoff point level , or “critical value”  Denoted: > =  > 1.96 = 0.025 . = 1.96  > 1.64 = 0.0505 . = 1.64  > 1.65 = 0.0495 . = 1.65  Keys: . = .; . = . PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 16 6.6. Binomial vs Normal  Binomial: ~(, ) with ≥ 100   approximate: (, )  With: = ; = (1− ) Example 6.3. Probability that visitor buy good in the shopping mall is 0.3. In 400 visitors, what is the probability  (a) There are at least 100 buyers  (b) Number of buyers is from 90 to 150 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 17 6.7. Cutoff Point  Normal Distribution:  Student Distribution:  df: Degree of freedom  Table 2 (p.976)  . = 1.833; . = 2.086  ≈  Chi-square Distribution:  Table 3 (p.979)  . = 3.94 ; . = 24.996 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 18 Key Concepts  Continuous variable  Density function  Normal distribution [1] Chapter 6:  (270) 3, 5  (281) 11, 12, 17, 19, 23, 24, 31  (292) 41, 44, 49 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 19 Exercise
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