ABSTRACT
In real-world problems we are often faced with one or more quantities that do not have fixed values. The values of such quantities depend on random actions, and they usually change from one experiment to another. For example, the number of babies born in a certain hospital each day is not a fixed quantity. It is a complicated function of many random factors that vary from one day to another. So are the following quantities: the arrival time of a bus at a station, the sum of the outcomes of two dice when thrown, the amount of rainfall in Seattle during a given year, the number of earthquakes that occur in California per month, and the weight of grains of wheat grown on a certain plot of land (it varies from one grain to another). In probability, quantities introduced in these diverse examples are called random variables. The numerical values of random variables are unknown. They depend on random elements occurring at the time of the experiment and over which we have no control. For example, if in rolling two fair dice, X is the sum, then X can only assume the values 2, 3, 4, … , $ \ldots , $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/inline-math4_1.tif"/> 12 with the following probabilities: P ( X = 2 ) = P ( { ( 1 , 1 ) } ) = 1 / 36 , P ( X = 3 ) = P ( { ( 1 , 2 ) , ( 2 , 1 ) } ) = 2 / 36 , P ( X = 4 ) = P ( { ( 1 , 3 ) , ( 2 , 2 ) , ( 3 , 1 ) } ) = 3 / 36 , $$ \begin{aligned} P(X=2)&=P\big (\big \{(1,1)\big \}\big ) = 1/36,\\ P(X=3)&=P\big (\big \{(1,2),(2,1)\big \}\big ) = 2/36,\\ P(X=4)&=P\big (\big \{(1,3),(2,2),(3,1)\big \}\big ) = 3/36, \end{aligned} $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429457951/63b50632-c513-448c-ba5f-4cb005c69347/content/um270.tif"/>
