Random Variables and Probability Distributions

1 POLI 270 - Mathematical and Statistical FoundationsProf. S. SaieghFall 2010 Lecture Notes - Class 8 November 18, Variables and Probability DistributionsWhen we perform an experiment we are often interested not in the particular outcomethat occurs, but rather in somenumberassociated with that example, in the game of craps a player is interested not in the particular numberson the two dice, but in theirsum. In tossing a coin 50 times, we may be interested only inthenumberof heads obtained, and not in the particular sequence of heads and tails thatconstitute the result of 50 both examples, we have a rule which assigns to each outcome of the experiment asingle real number.

2 Hence, we can say that afunctionis guys are already familiar with the function concept. Now we are going to look atsome functions that are particularly useful to study probabilistic/statistical VariablesIn Probability theory, certain functions of special interest are given special names:Definition 1A function whose domain is a sample space and whose range is some set ofreal numbers is called a Random variable . If the Random variable is denoted byXand hasthe sample space ={o1,o2,..,on}as domain, then we writeX(ok)for the value ofXatelementok.

3 ThusX(ok)is the real number that the function rule assigns to the elementokof .Lets look at some examples of Random Variables :Example 1 Let ={1,2,3,4,5,6}and defineXas follows:X(1) =X(2) =X(3) = 1,X(4) =X(5) =X(6) = a Random variable whose domain is the sample space and whose range isthe set{1, 1}.Xcan be interpreted as the gain of a player in a game in which a die isrolled, the player winning $1 if the outcome is 1,2,or 3 and losing $1 if the outcome is 4,5, 2 Two dice are rolled and we define the familiar sample space ={(1,1),(1,2),..(6,6)}containing 36 elements.

4 LetXdenote the Random variable whose value for any element of is the sum of the numbers on the two the range ofXis the set containing the 11 values ofX:2,3,4,5,6,7,8,9,10,11, ordered pair of has associated with it exactly one element of the range as requiredby Definition 1. But, in general, the same value ofXarises from many different exampleX(ok) = 5 is any one of the four elements of the event{(1,4),(2,3),(3,2),(4,1)}.Example 3A coin is tossed, and then tossed again. We define the sample space ={HH,HT,TH,TT}.IfXis the Random variable whose value for any element of is the number of headsobtained, thenX(HH) = 2, X(HT) =X(TH) = 1, X(TT) = that more than one Random variable can be defined on the same sample space.

5 Forexample, letYdenote the Random variable whose value for any element of is the numberof heads minus the number of tails. ThenX(HH) = 2, X(HT) =X(TH) = 0, X(TT) = now that a sample space ={o1,o2,..,on}is given, and that some acceptable assignment of probabilities has been made to the samplepoints in . Then ifXis a Random variable defined on , we can ask for the probabilitythat the value ofXis some number, event thatXhas the valuexis the subset of containing those elementsokforwhichX(ok) =x. If we denote byf(x) the Probability of this event, thenf(x) =P({ok |X(ok) =x}).

6 (1)Because this notation is cumbersome, we shall writef(x) =P(X=x),(2)adopting the shorthand X=x to denote the event written out in (1).2 Definition 2 The function f whose value for each real numberxis given by (2), or equiva-lently by (1), is called theprobability functionof the Random other words, the Probability function ofXhas the set of all real numbers as its domain,and the function assigns to each real numberxthe Probability thatXhas the 4 Continuing Example 1, if the die is fair, thenf(1) =P(X= 1) =12, f( 1) =P(X= 1) =12,and f(x) = 0ifxis different from 1 or 5If both dice in Example 2 are fair and the rolls are independent, so that eachsample point in has probability136, then we compute the value of the Probability functionatx= 5as follows:f(5) =P(X= 5) =P({(1,4),(2,3),(3,2),(4,1)}) = is the Probability that the sum of the numbers on the dice is 5.

7 We can computethe probabilitiesf(2),f(3),..,f(12) in an analogous values are summarized in the following table:x23456789101112f(x)136236336436536 636536436336236136 The table only includes those numbersxfor whichf(x)>0. And since we includeallsuch numbers, the probabilitiesf(x) in the table add to the Probability table of a Random variableX, we can tell at a glance not only thevarious values ofX, but also the Probability with which each value occurs. This informationcan also be presented graphically, as in the following is called theprobability chartof the Random variableX.

8 The various values ofXare indicated on the horizontalx-axis, and the length of the vertical line drawn from thex-axis to the point with coordinates (x,f(x)) is the Probability of the event thatXhas , we are often interested not in the Probability that the value of a Random variableXis a particular number, but rather in the Probability thatXhas some valueless than orequal tosome general, ifXis defined on the sample space , then the event thatXis less thanor equal to some number, sayx, is the subset of containing those elementsokfor whichX(ok) x.

9 If we denote byF(x) the Probability of this event (assuming an acceptableassignment of probabilities has been made to the sample points ), thenF(x) =P({ok |X(ok) x}).(3)In analogy with our argument in (2), we adopt the shorthand X x to denote theevent written out in (3), and then we can writeF(x) =P(X x).(4)Definition 3 The functionFwhose value for each real numberxis given by (4), or equiv-alently by (3), is called thedistribution functionof the Random other words, the distribution function ofXhas the set of all real numbers as its do-main, and the function assigns to each real numberxthe Probability thatXhas a value lessthan or equal to ( , at most)

10 The is an easy matter to calculate the values ofF, the distribution function of a randomvariableX, when one knowsf, the Probability function 6 Lets continue with the dice experiment of Example event symbolized byX 1 is the null event of the sample space , since the sum ofthe numbers on the dice cannot be at most 1. HenceF(1) =P(X 1) = eventX 2 is the subset{(1,1)}, which is the same as the eventX= 2. Thus,F(2) =P(X 2) =f(2) = eventX 3 is the subset{(1,1),(1,2),(2,1)}, which is seen to be the union of theeventsX= 2 andX= 3. Hence,F(3) =P(X 3) =P(X= 2) +P(X= 3)=f(2) +f(3)=136+236= , the eventX 4 is the union of the eventsX= 2,X= 3, andX= 4, so that136+236+336= this way, we obtain the entries in the followingdistribution tablefor therandom variableX:x23456789101112F(x)13633663610 361536213626363036333635363636 Remember, though, that the domain of the distribution functionFis the set ofallrealnumbers.