Econ 3120 1st Edition Lecture 1 Outline of Last Lecture I Syllabus review Outline of Current Lecture I Random Variables II Continuous Distributions III Multivariate Distributions Current Lecture 1 Random Variables A random variable is a variable that takes on numerical values and has an outcome that is determined by an experiment A discrete random variable takes on countably many values A continuous random variable takes on values that lie on a continuum e g an interval from 0 to 1 We denote the random variable by X or some other capital letter and the values the variable can take as x or some other lowercase letter The probability that a random variable X takes on a particular value x is denoted by P X x I 1 Discrete Distributions A random variable X has a discrete distribution or X is a discrete random variable if X can take only a finite number of values x1 x2 xkor at most a countably infinite sequence of values The probability distribution function of a discrete random variable X is defined as the function f x P X x and represents the probability that the random variable X takes on each potential value of x This is sometimes called the probability mass function but only for a discrete random variable Properties of f x for a discrete random variable 1 0 f x 1 for each value within the range of X 12 x f x 1 where the summation is over all values within the range of X The cumulative distribution function of a discrete random variable X is given by F x P X x t x f t x where f t is the probability distribution of X at point t Properties of F x 1 F 0 F 1 2 F x is nondecreasing in x i e if a b then F a F b 1 2 Continuous Distributions A random variable X has a continuous distribution or X is a continuous random variable if X can assume any value in an interval General rule With a continuous random variable we assign zero probability to single points In other words P X x 0 The probability density function for a continuous random variable X is defined as f x such that P a X b b a f x dx Unlike the case with discrete random variables the values of f x have no useful meaning note that f x 6 These notes represent a detailed interpretation of the professor s lecture GradeBuddy is best used as a supplement to your own notes not as a substitute P X x 0 Instead we assign probability to the integral of f x over a certain interval Properties of f x for a continuous random variable 1 f x 0 for x 2 f x dx 1 2 The cumulative distribution function for a continuous random variable whose pdf is f t is given by F x P X x x f t dt Properties of F x for a continuous random variable 1 F 0 F 1 2 F x is nondecreasing in x i e if a b then F a F b Other properties of the cdf 4 P a X b F b F a 5 P X a 1 P X a 1 F a 6 f x dF x dx where the derivative of F x exists Multivariate Distributions If we have two random variables X and Y we write the probability that X and Y take on particular values x and y as P X x Y y The joint probability distribution over X and Y is given by f x y For discrete random variables f x y represents P X x Y y For continuous random variables we have to define the probabilities over a range of X and Y similar to the univariate case P a X b c Y d b a d c f x y dy dx Properties of the joint distribution for random variables 1 f x y 0 for x y 2 x y f x y 1 discrete x y f x y dy dx 1 If X and Y are discrete random variables the joint cumulative distribution of X and Y is given by F x y P X x Y Y s x t y f s t for x y For a continuous random variable the joint cumulative distribution of X and Y is given by F x y P X x Y Y y x f s t ds dt for x y For continuous distributions we also have the following relationship between the joint cumulativeand the joint probability distributions density function f x y 2 x y F x y