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chap04PRN - Chapter 4.1 A random variable is a variable...

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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: ¾ 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 ¾ 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. ) x X ( 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: x 1 2 3 4 5 6 ) 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 ( P 0 d 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
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The cumulative probability function is defined as: for all possible values of a . ) a X ( P ) a ( F d This can be calculated from the probability distribution function as: ¦ d a x ) 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 ) 1 X ( P ) 1 X ( P ) 1 ( F d 3 1 6 1 6 1 ) 2 X ( P ) 1 X ( P ) 2 X ( P ) 2 ( F . . d 2 1 ) 3 X ( P ) 2 X ( P ) 1 X ( P ) 3 X ( P ) 3 ( F . . d 3 2 ) 4 X ( P ) 4 ( F d 6 5 ) 5 X ( P ) 5 ( F d 1 ) 6 X ( 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 0 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
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The cumulative probability function for a discrete random variable has the properties: x for all possible values of a . 1 ) a ( F 0 d d x For two numbers a , b with a < b then ) b ( F ) a ( F d x ) a ( F 1 ) a X ( P 1 ) a X ( P 0 d 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: x P(x) F(x) 0 0.10 0.10 1 0.08 0.18 2 0.07 0.25 3 0.15 0.40 4 0.12 0.52 5 0.08 0.60 6 0.10 0.70 7 0.12 0.82 8 0.08 0.90 9 0.10 1.00 Econ 325 – Chapter 4 5 What is the probability of five or more delayed flights in a given hour ?
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