The Practice of Statistics, d ed ates, Moore, ad Stares Itroductio We are ofte more iterested i the umber of times a give outcome ca occur tha i the possible outcomes themselves For example, if we toss a coi four times, we might be iterested i how may heads we come up with This leads us to a ew variable,, that ca take o ay of the 5 values 0, 1,, 3, or 4 Because each ru of this experimet (tossig a coi four times) could result i a differet value of, we call a radom variable A radom variable is a variable whose value is a umerical outcome of a radom pheomeo We are still iterested i assigig probabilities to the values of the radom variable, ad we will lear at least two ways of doig so 71 Discrete ad Cotiuous Radom Variables A discrete radom variable has a coutable umber of possible values We ca create a probability distributio of that lists the possible values ad their probabilities: Value of x 1 x x 3 x k Probability p 1 p p 3 p k Remember that a legitimate probability distributio has probabilities that must all: be umbers betwee 0 ad 1 add up to 1 We fid the probability of a evet by addig up the probabilities of the particular values of that make up the evet Study ad uderstad the probability histograms from pages 393 395 Whe we use the radom digit table, the outcomes are discrete, that is, there are a fiite umber of possible values (0 9) If we were iterested i a radom umber betwee 0 ad 1, the there would be a ifiite umber of possible values This latter case describes what is called a cotiuous radom variable A cotiuous radom variable takes all values i a iterval of umbers The probability distributio of is described by a desity curve Recall from chapter that a desity curve is always o or above the horizotal axis has area exactly 1 beeath it We fid the probability of a evet by calculatig the area uder the desity curve ad above the values of that make up the evet Page 1 of 5
The Practice of Statistics, d ed ates, Moore, ad Stares THIS IS A VER IMPORTANT DISTINCTION betwee probability models for discrete radom variables ad probability models for cotiuous radom variables: the probability model for cotiuous radom variables assigs probabilities to itervals of outcomes rather tha to idividual outcomes I fact, all cotiuous probability distributios assig probability 0 to every idividual outcome I practice, this meas that for a cotiuous radom variable, P ( 08) P ( 08) However, for a discrete radom variable, P ( > 08 ) ad P( 08) may have differet values > = KE POINT: Because a ormal desity curve is a desity curve, it is also a probability distributio 7 Meas ad Variaces of Radom Variables The mea still refers to the ordiary average However, with radom variables, ot all outcomes are ecessarily equally likely We take this ito cosideratio by calculatig the mea of a radom variable as a average weighted by its probability of occurrig Likewise, the variace is also a weighted average of squared deviatios of the outcomes of about the mea I geeral, the mea or expected value of a discrete radom variable with the probability distributio Value of x 1 x x 3 x Probability p 1 p p 3 p is give by the formula = x p = x p + x p + + x p i i 1 1 i= 1 This formula tells us to multiply each possible value of the radom variable by its correspodig probability of occurrig, ad add the results IMPORTANT: The mea of a radom variable is differet from the mea of a data set The mea of a data set is the mea of umbers that have already occurred, while the mea of a radom variable is the value you would expect if you were to perform the experimet several times I much the same way, if give the above distributio, we fid the variace of usig the formula = i i = + + + i= 1 ( x ) p ( x1 ) p1 ( x ) p ( x ) p σ This formula tells us to square the differece betwee each measuremet ad the mea ad the multiply the result by the correspodig probability of that outcome s occurrece Just as before, the stadard deviatio ( σ ) is simply the square root of the variace, or i symbols, σ Page of 5
The Practice of Statistics, d ed ates, Moore, ad Stares Law of Large Numbers As the umber of observatios draw at radom from ay populatio icreases, the mea x of the observed values evetually approaches the mea of the populatio (as closely as we may specify) ad the stays that close (or closer) This law holds for ay populatio (ot just those that are ormally distributed) Ufortuately, the law does ot tell us how may observatios make up a large umber of observatios This depeds o the variability of the outcomes! KE POINT: The Law of Large Numbers aswers the followig questio: If x is rarely exactly right (we would likely get a differet value of x every time we took a ew sample) why is it a reasoable estimate of the populatio mea? The mea of a radom variable is a mea i two seses 1 It is the average of the possible values, weighted by their probability of occurrig It is the log-ru average of may idepedet observatios o the variable Most people actually believe i a icorrect Law of Small Numbers, which demostrates their miscoceptios about radom behavior For example, most people would ot realize that the likelihood of flippig a coi ad receivig a ru of three or more heads is actually greater tha 80% Combiig Radom Variables I may cases it is ecessary to combie or trasform radom variables The basic rules for doig so are: RULES FOR MEANS 1 a+ b = a + b : Multiplyig each outcome by the same umber, addig the same umber to each outcome, or some combiatio of both, simply does the same thig to the mea of the origial radom variable ± ± : The mea of the sum (or differece) of two radom variables is the sum (or differece) of their meas RULES FOR VARIANCES 1 σ b σ a+ b = σ ± ± ρσ σ The Greek letter rho ( ρ ) is the correlatio betwee ad (recall this from chapter 3) If ad are correlated i ay way (ie, ot idepedet), there is some crossover i the variatio betwee the two variables, ad so we add (or subtract) a term that combies σ ad σ However, if ad are idepedet, the ρ = 0, ad the equatio becomes σ ± Page 3 of 5
The Practice of Statistics, d ed ates, Moore, ad Stares I always remember these rules by thikig of the hypothetical test example Suppose I give the class a 50 poit test The ask yourselves the followig questios (thik through the questios ad aswers util you are cofidet about their otatio ad iterpretatio): What happes to the mea ad stadard deviatio if I multiply every score by? The mea is also multiplied by Symbolically, The stadard deviatio is multiplied by two (ad the variace, sice it is the square of the stadard deviatio, is multiplied by 4) Symbolically, ( ) σ = σ or σ = σ What happes to the mea ad stadard deviatio if I the add 5 extra credit poits to everybody s test? The mea icreases by 5 Symbolically, 5+ = 5 + The stadard deviatio stays the same Symbolically, σ or σ 5+ 5+ Puttig the two above scearios together (doublig the scores ad addig 5 poits to each), we see that the mea is doubled ad the icreased by 5, ad the stadard deviatio is just doubled Now suppose I give two tests for chapter 10, ad the fial score will be the sum of those two tests This would create a ew radom variable + It makes sese that the average of the sum of ad would be the sum of the two averages, ie, + + The stadard deviatio is a little trickier, though To deal with the stadard deviatios i this particular case (or i ay case i which you have the sum or differece of two radom variables), it is crucial that you remember this sayig: Variaces ADD, but stadard deviatios DO NOT Traslatio: If you are give the stadard deviatios of ad, you MUST covert them to variaces (by squarig them), THEN add (or subtract) them, THEN covert the result back to a stadard deviatio (by takig the square root) I symbols, these last two rules follow: 1 ± ± σ ± ± ρσ σ If you are give variaces, the you ca just add them directly Oe more thig ote that it does t matter whether you are addig or subtractig the radom variables the VARIANCES always ADD The term that might ot add is the crossover term, ie, the ± ρσ σ term Ay liear combiatio of idepedet ormal radom variables is also ormally distributed I other words, if ad are idepedet ormal radom variables ad a ad b are ay fixed umbers, a + b is also ormally distributed The mea ad stadard deviatio of a + b is foud from the rules discussed above Page 4 of 5
The Practice of Statistics, d ed ates, Moore, ad Stares EAMPLES of combiig radom variables The followig example walks through icreasigly complicated algebraic combiatios of the radom variables ad/or Suppose for the followig problems that = 10, σ = ad = 1, σ = 3 Fid the meas, variaces, ad stadard deviatios of the followig ew radom variables New radom variable: (multiply a radom variable by a costat) Solutio: ( ) ( ) ( ) ( ) = = 10 = 0, σ = = 16 σ = 16 = 4 Explaatio: Double the mea The stadard deviatio also doubles, but sice we eed the variace, we square the doubled stadard deviatio Of course, oce we take the square root agai to get back to the stadard deviatio, we re back to just doublig the stadard deviatio I effect, if we are oly multiplyig a radom variable by some factor, (ad ot addig radom variables together), we ca simply multiply both the mea ad stadard deviatio by that factor (make sure the stadard deviatio is positive) New radom variable: + 3 (add a costat to a radom variable) + 3 = 10 + 3 = 13, σ = 4 σ = 4 = Solutio: ( ) + 3 + 3 + 3 Explaatio: Add 3 to the mea The stadard deviatio does ot chage New radom variable: 3 + (combie the previous two) Solutio: ( ) ( ) = 3 + = 38, σ = 3σ = 3 σ = 3 3 = 81 σ = 81 = 9 3 + 3 + 3 + Explaatio: Multiply the mea by 3, the add Multiplyig by 3 actually multiplies the variace by 9, ad hece the stadard deviatio by 3 Addig does ot affect the variace New radom variable: 4 3 (combie two of the first type above) Solutio: = 4 3 = 4, ( ) ( ) 4 3 ( ) ( ) 4 3 4 3 σ = 4σ + 3σ = 4 σ + 3 σ = 4 + 3 3 = 145 σ = 145 104 4 3 Explaatio: The ew mea is simply 4 times the mea of mius 3 times the mea of The stadard deviatio starts with realizig that we must covert to variaces ad add We multiply each variace by the square of the correspodig coefficiet ad add the results Fially, we take the square root to get back to a stadard deviatio New radom variable: + (with radom variables, this is ot the same as ) Solutio: (compare to the case above ad make sure you uderstad the differece!) + + = 0, σ + = + = 8 σ + = 8 88 Explaatio: Add the mea of to itself Add the variace to itself, the take the square root to obtai the stadard deviatio The mea is the same as it was for the variable, but ote that the variace ad hece stadard deviatio are differet! Page 5 of 5