This preview shows pages 1–5. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Chapter 4.1 Random Variables A random variable is a variable that takes on numerical outcomes defined over a sample space of a random experiment. A random variable has a probability distribution. A random variable can be denoted by X (uppercase) and a possible numerical outcome is x (lowercase). Example: The random variable X is age in years of a UBC student. The possible outcomes are: x = 15, 16, 17, 18, . . . etc. Types of random variables: A discrete random variable has a countable number of values (typically integer numbers). Example 1: age in years of a UBC student Example 2: categorical variables. For example, the random variable X represents gender. The possible values can be assigned the codes: x = 0 male x = 1 female A continuous random variable can take any numerical value in an interval of the real number line. Examples: income, stock market prices, interest rates, consumer price index, etc. Econ 325 Chapter 4 1 Chapter 4.2 Discrete Random Variables For a discrete random variable X the probability distribution function is: for all possible values of x. )xX(P)x(P Example: The random variable X is the number resulting from the throw of a sixsided dice. The probability distribution function is: x123456 )x(P 6 1 6 1 6 1 6 1 6 1 6 1 That is, 6 1 )x( P for x = 1, 2, 3, 4, 5, 6 The probability distribution function for a discrete random variable has the properties: x 1)x(P0 d for all possible values of x. x 1)x(P x n summation over all possible values of x. Econ 325 Chapter 4 2 The cumulative probability function is defined as: for all possible values of a . )aX(P)a(F d This can be calculated from the probability distribution function as: d ax )x(P )a(F n summation over all possible values of x that are less than or equal to a . Example: For the dice throwing experiment: 6 1 )1X(P)1X(P)1(F d 3 1 6 1 6 1 )2 X(P)1X(P )2X(P)2(F . d 2 1 )3 X(P)2 X(P)1X(P )3X(P)3(F d 3 2 )4 X(P)4(F d 6 5 )5X(P)5(F d 1)6X(P)6(F d Econ 325 Chapter 4 3 Graph of the cumulative probability function for the dice throwing experiment. 1 5/6 1/6 6 5 4 3 2 1 F(x) x The graph illustrates that, for a discrete random variable, the cumulative probability function is a step function that begins at 0 and ends at 1. Econ 325 Chapter 4 4 The cumulative probability function for a discrete random variable has the properties: x for all possible values of a . 1)a(F0 d d x For two numbers a , b with a < b then )b(F)a(F d x )a(F1 )aX(P1 )aX(P 0 0 " Example: Exercise 4.14, page 142. The random variable X is the number of flights delayed per hour at an international airport. The probability distribution function and cumulative probability function are: xP(x)F(x) 00.100.10 10.080.18 20.070.25 30.150.40 40.120.52 50.080.60 60.100.70 70.120.82 80.080.90 90.101.00 Econ 325 Chapter 4 5 What is the probability of five or more delayed flights in a given hour ? 48.0 52.0 1 )4(F1 )4 X(P1 )5X(P 0 0 t What is the probability of three through seven (inclusive) delayed flights in a given hour ? 57.0 25.0 82.0 )2(F)7(F )2X(P)7X(P )7 X3(P d d Econ 325 Chapter 4 6 Chapter 4.3 Chapter 4....
View
Full
Document
 Spring '10
 WHISTLER
 Economics

Click to edit the document details