Empirical Distributions

Transcription

1 Emirical Distributios A emirical distributio is oe for which each ossible evet is assiged a robability derived from exerimetal observatio. It is assumed that the evets are ideedet ad the sum of the robabilities is. A emirical distributio may rereset either a cotiuous or a discrete distributio. If it reresets a discrete distributio, the samlig is doe o ste. If it reresets a cotiuous distributio, the samlig is doe via iterolatio. The way the table is described usually determies if a emirical distributio is to be hadled discretely or cotiuously; e.g., discrete descritio cotiuous descritio value robability value robability To use liear iterolatio for cotiuous samlig, the discrete oits o the ed of each ste eed to be coected by lie segmets. This is rereseted i the grah below by the gree lie segmets. The stes are rereseted i blue: rsamle x.5 I the discrete case, samlig o ste is accomlished by accumulatig robabilities from the origial table; e.g., for x =.4, accumulate robabilities util the cumulative robability exceeds.4; rsamle is the evet value at the oit this haes (i.e., the cumulative robability is the first to exceed.4, so the rsamle value is 35). I the cotiuous case, the ed oits of the robability accumulatio are eeded, i this case x=.5 ad x=.65 which rereset the oits (.5,) ad (.65,35) o the grah. From basic college algebra, the sloe of the lie segmet is (35-)/(.65-.5) = 5/.4 = The sloe = 37.5 = (35-rsamle)/(.65-.4) so rsamle = 35 - (37.5.5) = = 5.65.

2 Discrete Distributios To ut a little historical ersective behid the ames used with these distributios, James Beroulli (654-75) was a Swiss mathematicia whose book Ars Cojectadi (ublished osthumously i 73) was the first sigificat book o robability; it gathered together the ideas o coutig, ad amog other thigs rovided a roof of the biomial theorem. Siméo-Deis Poisso (78-84) was a rofessor of mathematics at the Faculté des Scieces whose 837 text Recherchés sur la robabilité des jugemets e matière crimielle et e matière civile itroduced the discrete distributio ow call the Poisso distributio. Kee i mid that scholars such as these evolved their theories with the objective of rovidig sohisticated abstract models of real-world heomea (a effort which, amog other thigs, gave birth to the calculus as a major modelig tool). I. Beroulli Distributio A Beroulli evet is oe for which the robability the evet occurs is ad the robability the evet does ot occur is -; i.e., the evet is has two ossible outcomes (usually viewed as success or failure) occurrig with robability ad -, resectively. A Beroulli trial is a istatiatio of a Beroulli evet. So log as the robability of success or failure remais the same from trial to trial (i.e., each trial is ideedet of the others), a sequece of Beroulli trials is called a Beroulli rocess. Amog other coclusios that could be reached, this meas that for trials, the robability of successes is. A Beroulli distributio is the air of robabilities of a Beroulli evet, which is too simle to be iterestig. However, it is imlicitly used i yeso decisio rocesses where the choice occurs with the same robability from trial to trial (e.g., the customer chooses to go dow aisle with robability ) ad ca be case i the same kid of mathematical otatio used to describe more comlex distributios: () = (-) - for =, otherwise () The exected value of the distributio is give by E(X) = (-) + = - The stadard deviatio is give by (-) While this is otatioal overkill for such a simle distributio, it s costructio i this form will be useful for uderstadig other distributios.

3 Samlig from a discrete distributio, requires a fuctio that corresods to the distributio fuctio of a cotiuous distributio f give by F(x) = This is give by the mass fuctio F(x) of the distributio, which is the ste fuctio obtaied from the cumulative (discrete) distributio give by the sequece of artial sums x () x f()d For the Beroulli distributio, F(x) has the costructio for - x < F(x) = - for x < for x which is a icreasig fuctio (a so ca be iverted i the same maer as for cotiuous distributios). Grahically, F(x) looks like F(x) - which iverted yields the samlig fuctio rsamle - x I other words, for radom value x draw from [,), rsamle = if x < - if - x < I essece, this demostrates that samlig from a discrete distributio, eve oe as simle as the Beroulli distributio, ca be viewed i the same maer as for cotiuous distributios.

4 II. Biomial Distributio The Beroulli distributio reresets the success or failure of a sigle Beroulli trial. The Biomial Distributio reresets the umber of successes ad failures i ideedet Beroulli trials for some give value of. For examle, if a maufactured item is defective with robability, the the biomial distributio reresets the umber of successes ad failures i a lot of items. I articular, samlig from this distributio gives a cout of the umber of defective items i a samle lot. Aother examle is the umber of heads obtaied i tossig a coi times. The biomial distributio gets its ame from the biomial theorem which states that the biomial k -k! ( a + b) = a b where = k k k!( - k)! It is worth oitig out that if a = b =, this becomes ( + ) = = k Yet aother viewoit is that if S is a set of sie, the umber of k elemet subsets of S is give by! = k!( - k)! k This formula is the result of a simle coutig aalysis: there are! ( -)... ( - k + ) = ( - k)! ordered ways to select k elemets from ( ways to choose the st item, (-) the d, ad so o). Ay give selectio is a ermutatio of its k elemets, so the uderlyig subset is couted k! times. Dividig by k! elimiates the dulicates. Note that the exressio for couts the total umber of subsets of a -elemet set. For ideedet Beroulli trials the df of the biomial distributio is give by () = ( ) otherwise for =,,..., Note that by the biomial theorem, () = ( + (- )) =, verifyig that () is a df.

5 Whe choosig items from amog items, the term ( ) reresets the robability that are defective (ad cocomitatly that (-) are ot defective). The biomial theorem is also the key for determiig the exected value E(X) for the radom variable X for the distributio. E(X) is give by E(X) = ( ) i i (the exected value is just the sum of the discrete items weighted by their robabilities, which corresods to a samle s mea value; this is a extesio of the simle average value obtaied by dividig by, which corresods to a weighted sum with each item havig robability /). For the biomial distributio the calculatio of E(X) is accomlished by! term i commo - - E(X) = ( ) = ( )!( - )! (- ) = (- ) - = = ( + - ) = reset i every summad aly the biomial theorem to this This gives the result that E(X) = for a biomial distributio o items where robability of success is. It ca be show that the stadard deviatio is ( ) The biomial distributio with = ad =.7 aears as follows: ().3 mea

6 Its corresodig mass fuctio F(x) is give by F() which rovides the samlig fuctio rsamle x A tyical samlig tactic is to accumulate the sum rsamle () icreasig rsamle util the sum's value exceeds the radom value betwee ad draw for x. The fial rsamle summatio limit is the samle value.

7 III. Poisso Distributio (values =,,,...) The Poisso distributio is the limitig case of the biomial distributio where ad. The exected value E(X) = λ where λ as ad. The stadard deviatio is λ. The df is give by λ λ e () =! This distributio dates back to Poisso's 837 text regardig civil ad crimial matters, i effect scotchig the tale that its first use was for modelig deaths from the kicks of horses i the Prussia army. I additio to modelig the umber of arrivals over some iterval of time (recall the relatioshi to the exoetial distributio; a Poisso rocess has exoetially distributed iterarrival times), the distributio has also bee used to model the umber of defects o a maufactured article. I geeral the Poisso distributio is used for situatios where the robability of a evet occurrig is very small, but the umber of trials is very large (so the evet is exected to actually occur a few times). Grahically, with λ =, it aears as: () mea The samlig fuctio looks like: rsamle x

8 IV. Geometric Distributio The geometric distributio gets its ame from the geometric series: < r, r =, r =, - r (- r) for r ( + ) r = (- r) various flavors of the geometric series The df for the geometric distributio is give by - ( ) for =,,... () = otherwise The geometric distributio is the discrete aalog of the exoetial distributio. Like the exoetial distributio, it is "memoryless" (ad is the oly discrete distributio with this roerty; see the discussio of the exoetial distributio). Its exected value is give by E(X) = (- ) - = (-+ ) (by alyig the 3 rd form of the geometric series). The stadard deviatio is give by. A lot of the geometric distributio with =.3 is give by = () mea A tyical use of the geometric distributio is for modelig the umber of failures before the first success i a sequece of ideedet Beroulli trials. This is a tyical sceario for sales. Suose that the robability of makig a sale is.3. The () =.3 is the robability of success o the st try, () = (-) =.7.3 =. which is the robability of failig o the st try (with robability -) ad succeedig o the d (with robability ). (3) = (-)(-) =.5 is the robability that the sale takes 3 tries, ad so forth. A radom samle from the distributio reresets the umber of attemts eeded to make the sale.