Lecture Probability & statistics: Chapter 5 - Bùi Dương Hải

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Lecture "Probability & statistics - Chapter 5: Discrete probability" has contents: Random variable, probability distribution, expected value, variance – standard deviation, bivariate probability, binomial distribution.

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Nội dung Text: Lecture Probability & statistics: Chapter 5 - Bùi Dương Hải

Lecture 5. DISCRETE PROBABILITY  Random Variable  Probability Distribution  Expected value  Variance – Standard Deviation  Bivariate Probability  Binomial Distribution  [1] Chapter 5: pp.215 - 260 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 1
5.1. Random Variable  Random variable: numerical value from a random experiment.  Denoted by X, Y, Z, or X1, X2,...  Ex. Tossing a die, X is the number of dots - Number of boys in a 3-children family - Score of students’ exam - Temparature during a day - Interest rates in a period of time PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 2
Types of Random Variable  Variable , value is random  Discrete variable: = ( , , … , )  Number of item: = (0, 1, 2, … )  Score of test: = (0, 1, 2, … , 100)  ( = ) is a random event  Continuous variable: =( ; )  Time  Temperature  Length, Weight PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 3
5.2. Discrete Probability Distribution  Discrete: = ( , ,…, )  Denote: = = Value … Probability …  Property: ∑ =1  is discrete probabitily distribution; probabitiy function PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 4
Example Ex. Probability distribution of X, which is the number of Heads when flipping a coin twice Flip a coin twice X = {0, 1, 2} p x 0 1 2 P(x) 1/4 2/4 1/4 Example 5.1. Number of Head when flipping a coin 3 times X 0 1 2 3 Probability PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 5
5.3. Parameter  Parameter of Random variable:  Expected value (Mean)  Variance, Standard Deviation  Ex. Salary ($) 7 8 9 Frequency 2 5 3 Percent 20% 50% 30% Probability 0.2 0.5 0.3 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 6
Expected Value  Expected value of X, denoted by E(X) or μX = =∑  Expected value of X is also Population Mean, and has the same unit with X.  Properties: if is a constant ( ) = ( + ) = ( )+ ( ) = ( ) ( ± ) = ( )± ( ) PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 7
Variance – Standard Deviation  Variance of is denoted by ( ) or ( ) or = − =∑ − Unit of Variance is square of unit of  Standard Deviation of is denoted by ( ) or = ( ) Unit of Variance is unit of PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 8
Comparison Example 5.1. Compare return rate of three projects Project Return rate (%) 7 A Probability 1 Project Return rate (%) 5 15 B Probability 0.5 0.5 Project Return rate (%) –10 10 24 C Probability 0.2 0.3 0.5 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 9
Properties of E(X) and V(X)  , are variable; is constant Expected Value Variance = =0 + = + + = × = × × = × ± = ± ( ) ± = + ±2 ( , ) ± = + If X and Y are independent PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 10
Investment Example 5.3. There are 4 independent projects, each have the same return rate probability distribution: Return rate (%) 0 20 Probability 0.3 0.7  Expected value and Variance when: (a) Invest 10 ($ mil.) in one project (b) Invest 40 ($ mil.) in one project (c) Invest in 4 projects, each 10 ($ mil.) PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 11
5.4. Bivariate Probability Example 5.5. Profit of Project 1 and 2 are and , respectively, with Bivariate Probability table: X Y –1 0 5  –2 0.05 0.1 0.05 0.2 7 0.05 0.2 0.55 0.8  0.1 0.3 0.6 1  Fill the blanks  , , , , + , ( + )? PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 12
Covariance and Correlation  Covariance , = − =∑ ∑ . . − ( )  ± = + ±2 ( , )  ± = + ±2 ( , )  Correlation , , = PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 13
Porfolio  Ex. Two investment projects A and B Project A B % for A % for B ( ) ( ) E(return) 10 20 100% 0% 10 5.00 90% 10% 11 4.54 (return) 5 12 80% 20% 12 4.45 70% 30% 13 4.76  = −6 60% 40% 14 5.40 50% 50% 15 6.26 40% 60% 16 7.28 30% 70% 17 8.38 20% 80% 18 9.55 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 14
5.5. Binomial Distribution  Bernoulli problem: independent experiments, probability of .  is number of success  Distribution of X is Binomial: ~ ( , ) = , = −  =  = − ; = ( − ) PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 15
Binomial Distribution Binomial Table (Table) n x P … .20 … 3 0 .5120 1 .3840 2 .0960 3 .0080 Ex. ( = 1| = 3, = 0.2) = 0.384 ( = 6| = 10, = 0.3) = 0.1029 ( = 4| = 10, = 0.7) = 0.1029 PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 16
Example Example 5.6. The quiz includes 10 multiple choices questions, each has 4 options and only one correct. A candidate do all questions by random choose the answers. (a) Expectation and variance of number of correct answer? (b) Probability that there are 3 correct answers? (c) Probability that there are at least 6 correct ones? (d) Each correct one is evaluated (+4) points, but for incorrect one, it is (-1) point. What is the chance for candidate gain 10 points in total ? PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 17
5.6. Poisson Distribution  Denoted: ~ ( )  = = 0,1,2 … !  = ; =  Binomial Distribution with large and small (that (1 – )) converges to Poisson Distribution, with l = . Ex. The number of mistake papers of a photo machine in one day is Poisson distribution with mean of 3. Find the probability that in the following day, there will be 4 mistake papers PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 18
Poisson Distribution – Table x Ex. ~ ( = 3) … 3.0 … 0 .0498 =4 =3 = = 1 .1494 ! 2 .2240 3 .2240 Using Table 7 (p.995), = 4 .1680 5 .1008 =4 =3 = 6 .0504 … … Example 5.7. The probability that a passenger forgets his (her) luggage on train is 0.008. What the probability that in 400 passengers, there is (a) No forgotten luggage (b) At least 4 forgotten luggages PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 19
Key Concepts  Random Variable  Discrete Variable  Probability Distribution  Expected Value  Variance, Standard Deviation  Bivariate Probability Distribution  Covariance  Binomial Distribution, Poisson Distribution PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 20