{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

chap04PRN

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

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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 ?

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern