chap04PRN

chap04PRN - Chapter 4.1 Random Variables A random variable...

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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 (upper-case) and a possible numerical outcome is x (lower-case). 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 six-sided 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....
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chap04PRN - Chapter 4.1 Random Variables A random variable...

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