Ch. 2-3 Review of Probability and Statistics

Ch. 2-3 Review of Probability and Statistics - 2. REVIEW of...

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1 | Reviews 2. REVIEW of PROBABILITY AND STATISTICS (CHAP. 2 - 3) [1] Important Concepts and Formulas (1) Population : The group of interest. EX: Household incomes of Phoenix residents, US GNP. (2) Probability (or frequency): A small island with 12 households Income (per day) # of households Probability $100 2 2/12 = 1/6 $200 2 2/12 = 1/6 $300 4 4/12 = 1/3 $400 4 4/12 = 1/3 ---------------------------------------------- 1 2 1
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2 | Reviews • Let X = a household's income ( X : random variable ). Describe the probability that X takes a specific value by: f ( x ) = 1/6, if x = 100 or 200; f ( x ) = 1/3 if x = 300 or 400. ( probability density function .) (3) Expected value (Population mean) of X : In the above example, the population mean is (100×2+200×2+300×4+400×4)/12 = 283.3 Expected value of X : () Xx E xx f x  E x = 100×(1/6) + 200×(1/6) + 300×(1/3) + 400×(1/3) = 283.3. Lesson: If you know the value of f ( x ) for each possible value of x , can compute the population mean.
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3 | Reviews (4) Population Variance of X Wish to know the dispersion of a population of size B : Let x 1 , . .. , x B be all members of population. Use 2 1 1 () ( ) B ii x var x x B  . • Alternatively, 22 2 2 var( ) ( ) ( ) ( ) x xx x x f x x f x  . In the above example, 12 2 1 var( ) ( ) /12 x = {(100-283.3) 2 + (100-283.3) 2 + (200-283.3) 2 + (200-283.3) 2 + (300-283.3) 2 + (300-283.3) 2 + (300-283.3) 2 + (300-283.3) 2 + (400-283.3) 2 + (400-283.3) 2 + (400-283.3) 2 + (400-283.3) 2 }/12 = 11388.889. var( x ) = Σ x ( x - μ x ) 2 f ( x ) = (100-283.3) 2 ×(1/6) + (200-283.3) 2 ×(1/6) + (300-283.3) 2 ×(1/3) + (400-283.3) 2 ×(1/3) = 11388.889.
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4 | Reviews Finance Question. X = Monthly return rate of Intel stock. Why do you want to know () Ex and var( ) x ? How would you make an investment decision? Suppose: ( ) 0.01(1%) & ( ) var( ) 0.3 (30%) s ex x  . monthly interest rate from a saving account = 0.005 (0.5%). What if monthly interest rate = 0%? What if ( ) 1(100%) ? What if ( ) 0 sex ?
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5 | Reviews (5) Case of Two Random Variables EX: Income ( X ) and consumption ( Y ) of the 12 households. Y ($)\ X ($) 100 200 300 400 30 1 1 2 1 : 5 40 1 0 1 1 : 3 50 0 1 1 2 : 4 ---------------------------------------------------------------------- 2 2 4 4 : 12
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6 | Reviews 1. Joint Probability Density Function Y\X 100 200 300 400 30 1/12 1/12 2/12 1/12 : 5/12 40 1/12 0 1/12 1/12 : 3/12 50 0 1/12 1/12 2/12 : 4/12 ------------------------------------------------------- 2/12 2/12 4/12 4/12 : 1 Joint pdf = f ( x , y ): f (100,50) = 0 1/12 = 8.3%.
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7 | Reviews 2. Marginal PDFs of X and Y : Marginal pdf of X = f x ( x ) = Σ y f ( x , y ) = Pr( X = x ) regardless of Y . [Unconditional = marginal] Marginal pdf of Y = f y ( y ) = Σ x f ( x,y ) = Pr( Y = y ) regardless of X . Y\X 100 200 300 400 f y ( y ) 30 1/12 1/12 2/12 1/12 : 5/12 40 1/12 0 1/12 1/12 : 3/12 50 0 1/12 1/12 2/12 : 4/12 ------------------------------------------------------- f x ( x ) 2/12 2/12 4/12 4/12 : 1
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8 | Reviews 3. Conditional pdf : ( | ) fy x = Pr( Y = y , given X = x ) = (, ) () x f x y f x ; ( | ) f x y = Pr( X = x , given Y = y ) = y f x y . (3 0 | 100) x  = (100,30) 1/12 1 (100) 2 /12 2 x f f ; (4 0 | 300) x = (300,40) 1 (300) 4 /12 4 x f f . 4. Population means and variances of X and Y xx E f x  ; yy E f y   ; 2 var( ) ( ) ( ) x f x ; 2 var( ) ( ) ( ) y yyf y 
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9 | Reviews 5. Conditional means and conditional variances (|) xx E xy x f xy  ; y E yx y fyx   ; 2 v a r [ ] x x Exy fxy ; 2 v a r [ ] (|) y y Eyx fyx . EX: ( | 200) ( | 200) (200,30) (200,40) (200,50) 30 40 50 (200) (200) (200) 1/12 0 30 40 50 40 2 /12 y xxx Ey x y f y x fff    2 222 var( | 200) ( ( | 200)) ( | 200) 0 (30 40) (40 40) (50 40) 100 2/12 y y Eyx  
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