## Random Variables

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Introduction

- Lecture number:
- 1
- Pages:
- 2
- Type:
- Lecture Note
- School:
- Cornell University
- Course:
- Econ 3120 - Applied Econometrics
- Edition:
- 1

**Unformatted text preview: **

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

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