Review - Mathematical Tools & Probability Logarithm - PowerPoint PPT Presentation

Mathematical Tools Summation Operator The Natural Review - Mathematical Tools & Probability Logarithm Fundamentals of Probability Discrete & Continuous Random Variable Caio Vigo Features of Probability Distributions Expected Value Variance The University of Kansas Standard Deviation Department of Economics Covariance Conditional Expectation Distributions Fall 2019 These slides were based on Introductory Econometrics by Jeffrey M. Wooldridge (2015) 1 / 50

Topics Mathematical Tools 1 Mathematical Tools Summation Operator The Natural Summation Operator Logarithm The Natural Logarithm Fundamentals of Probability Discrete & 2 Fundamentals of Probability Continuous Random Variable Features of Discrete & Continuous Random Variable Probability Distributions Features of Probability Distributions Expected Value Variance Expected Value Standard Deviation Covariance Variance Conditional Expectation Standard Deviation Distributions Covariance Conditional Expectation Distributions 2 / 50

Summation Operator Mathematical Tools Summation Operator The Natural Logarithm Fundamentals of Probability Discrete & It is a shorthand for manipulating expressions involving sums. Continuous Random Variable Features of n Probability Distributions � x i = x 1 + x 2 + . . . + x n Expected Value Variance i =1 Standard Deviation Covariance Conditional Expectation Distributions 3 / 50

Summation Operator - Properties Mathematical Tools Summation Operator Property 1: For any constant c , The Natural Logarithm Fundamentals n of Probability � c = nc Discrete & Continuous Random Variable i =1 Features of Probability Distributions Expected Value Variance Standard Deviation Property 2: For any constant c , Covariance Conditional Expectation n n Distributions � � cx i = c x i i =1 i =1 4 / 50

Summation Operator - Properties Mathematical Tools Property 3: If { ( x 1 , y 1 ) , ( x 2 , y 2 ) , . . . , ( x n , y n ) } is a set of n pairs of numbers, and Summation Operator The Natural Logarithm a and b are constants, then: Fundamentals of Probability n n n Discrete & � � � ( ax i + by i ) = a x i + b y i Continuous Random Variable Features of i =1 i =1 i =1 Probability Distributions Expected Value Variance Standard Deviation Average Covariance Conditional Expectation Given n numbers { x 1 , x 2 , . . . , x n } , their average or (sample) mean is given by: Distributions n x = 1 � ¯ x i n i =1 5 / 50

Summation Operator - Properties Mathematical Tools Property 4: The sum of deviations from the average is always equal to 0 , i.e.: Summation Operator The Natural Logarithm n Fundamentals � ( x i − ¯ x ) = 0 of Probability Discrete & i =1 Continuous Random Variable Features of Probability Distributions Property 5: Expected Value n n Variance x ) 2 = � � Standard Deviation ( x i − ¯ x i ( x i − ¯ x ) Covariance Conditional i =1 i =1 Expectation Distributions Property 6: � n = � n i =1 ( x i − ¯ x )( y i − ¯ y ) i =1 x i ( y i − ¯ y ) = � n i =1 y i ( x i − ¯ x ) 6 / 50

Summation Operator - Properties Mathematical Tools Summation Operator The Natural Common Mistakes Logarithm Fundamentals Notice that the following does not hold: of Probability Discrete & Continuous Random n � n x i i =1 x i Variable � � = Features of � n Probability y i i =1 y i Distributions i =1 Expected Value Variance Standard Deviation Covariance Conditional Expectation � n � 2 n Distributions � x 2 � i � = x i i =1 i =1 7 / 50

The Natural Logarithm Mathematical Tools Summation Operator The Natural Logarithm Fundamentals of Probability • Most important nonlinear function in econometrics Discrete & Continuous Random Variable Natural Logarithm Features of Probability Distributions y = log ( x ) Expected Value Variance Standard Deviation Other notations: ln ( x ) , log e ( x ) Covariance Conditional Expectation Distributions 8 / 50

The Natural Logarithm Mathematical Tools Figure: Graph of y = log ( x ) Summation Operator The Natural Logarithm Fundamentals of Probability Discrete & Continuous Random Variable Features of Probability Distributions Expected Value Variance Standard Deviation Covariance Conditional Expectation Distributions Source: Wooldridge, Jeffrey M. (2015). Introductory Econometrics: A Modern Approach. 9 / 50

The Exponential Function Mathematical Tools exp (0) = 1 Summation Operator The Natural exp (1) = 2 . 7183 Logarithm Fundamentals of Probability Figure: Graph of y = exp ( x ) (or y = e x ) Discrete & Continuous Random Variable Features of Probability Distributions Expected Value Variance Standard Deviation Covariance Conditional Expectation Distributions Source: Wooldridge, Jeffrey M. (2015). Introductory Econometrics: A Modern Approach. 10 / 50

The Natural Logarithm Mathematical Tools • Things to know about the Natural Logarithm y = log ( x ) : Summation Operator The Natural Logarithm • is defined only for x > 0 Fundamentals • the relationship between y and x displays diminishing marginal returns of Probability Discrete & Continuous Random • log ( x ) < 0 , for 0 < x < 1 Variable Features of Probability • log ( x ) > 0 , for x > 1 Distributions Expected Value • log (1) = 0 Variance Standard Deviation • Property 1: log ( x 1 x 2 ) = log ( x 1 ) + log ( x 2 ) , x 1 , x 2 > 0 Covariance Conditional Expectation • Property 2: log ( x 1 /x 2 ) = log ( x 1 ) − log ( x 2 ) , x 1 , x 2 > 0 Distributions • Property 3: log ( x c ) = c.log ( x ) , for any c • Approximation: log (1 + x ) ≈ x , for x ≈ 0 11 / 50

Topics Mathematical Tools 1 Mathematical Tools Summation Operator The Natural Summation Operator Logarithm The Natural Logarithm Fundamentals of Probability Discrete & 2 Fundamentals of Probability Continuous Random Variable Features of Discrete & Continuous Random Variable Probability Distributions Features of Probability Distributions Expected Value Variance Expected Value Standard Deviation Covariance Variance Conditional Expectation Standard Deviation Distributions Covariance Conditional Expectation Distributions 12 / 50

Random Variable Mathematical Tools Summation Operator The Natural Logarithm Fundamentals • A random variable (r.v.) is one that takes on numerical values and has an of Probability outcome that is determined by an experiment. Discrete & Continuous Random Variable Features of Probability • Precisely, an r.v. is a function of a sample space Ω to the Real numbers. Distributions Expected Value Variance Standard Deviation • Points ω in Ω are called sample outcomes, realizations, or elements . Covariance Conditional Expectation • Subsets of Ω are called events . Distributions 13 / 50

Random Variable Mathematical Tools Summation Operator The Natural Logarithm • Therefore, X is a r.v. if X : Ω → R Fundamentals of Probability Discrete & Continuous Random • Random variables are always defined to take on numerical values, even when they Variable Features of describe qualitative events. Probability Distributions Expected Value Variance Standard Deviation Example Covariance Conditional Expectation • Flip a coin, where Ω = { head, tail } Distributions 14 / 50

Discrete Random Variable Mathematical Tools Summation Operator The Natural Logarithm Fundamentals of Probability Probability Function Discrete & Continuous Random Variable X is a discrete r.v. if takes on only a finite or countably infinite number of values. Features of Probability Distributions Expected Value We define the probability function or probability mass function for X by Variance Standard Deviation f X ( x ) = P ( X = x ) Covariance Conditional Expectation Distributions 15 / 50

Continuous Random Variable Mathematical Tools Probability Density Function (pdf) Summation Operator The Natural Logarithm • A random variable X is continuous if there exists a function f X such that � ∞ Fundamentals f X ( x ) ≥ 0 for all x , −∞ f X ( x ) dx = 1 and for every a ≤ b , of Probability Discrete & Continuous Random � b Variable Features of P ( a < X < b ) = f X ( x ) dx Probability Distributions a Expected Value Variance Standard Deviation The function f X is called the probability density function (pdf). We have that Covariance Conditional Expectation � x Distributions F X ( x ) = f X ( t ) dt −∞ and f X ( x ) = F ′ X ( x ) at all points x at which F X is differentiable. 16 / 50

Joint Distributions and Independence Mathematical Tools Summation Operator The Natural Logarithm Fundamentals of Probability • We are usually interested in the occurrence of events involving more than one r.v. Discrete & Continuous Random Variable Features of Probability Example Distributions Expected Value • Conditional on a person being a student at KU, what is the probability that s/he Variance Standard Deviation attended at least one basketball game in Allen Fieldhouse? Covariance Conditional Expectation Distributions 17 / 50