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## Applied Statistics

by: Theodora Daniel I

17

0

19

# Applied Statistics CSE 2400

Theodora Daniel I
Florida Tech
GPA 3.9

Gerald Marin

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COURSE
PROF.
Gerald Marin
TYPE
Class Notes
PAGES
19
WORDS
KARMA
25 ?

## Popular in ComputerScienence

This 19 page Class Notes was uploaded by Theodora Daniel I on Monday October 12, 2015. The Class Notes belongs to CSE 2400 at Florida Institute of Technology taught by Gerald Marin in Fall. Since its upload, it has received 17 views. For similar materials see /class/221677/cse-2400-florida-institute-of-technology in ComputerScienence at Florida Institute of Technology.

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Date Created: 10/12/15
The Mathcad Advisor Probability Distributions Mathcad has 17 builtin probability distributions including The Beta Distribution The Binomial Distribution The Cauchy Distribution The Chisquared Distribution The Exponential Distribution The F Distribution The Gamma Distribution The Geometric Distribution 9 The Hypergeometric Distribution 10 The Logistic Distribution 11 The Lognormal Distribution 12 The Negative Binomial Distribution 13 The Normal Distribution 14 The Poisson Distribution 15 The Student t Distribution 16 The Uniform Distribution 17 The Weibull Distribution WNO U39IwaH Each distribution has four functions associated with it I A probability density function d followed by the distribution name In continuous distributions the probability density is the likelihoodperunitx that a random variable will take on a particular value within a particular distribution The area under this curve between two values of x corresponds to the probability of having a future measurement in the given distribution fall between these values Note that in the limit as the ends of the integration interval converge the probability is 0 of achieving a particular value since the area under the curve drops to zero In discrete distributions the probability density is simply the likelihood that a random variable will take a particular value I A cumulative probability distribution function p followed by the distribution name The probability that a random variable will take on a value less than or equal to a specified value This is obtained by simply integrating or summing for a discrete distribution the corresponding probability density over an appropriate range I An inverse cumulative probability distribution function q followed by the function name These functions take a probability as an argument and return a value such that the probability that a random variable will be less than or equal to that value is whatever probability you supplied as an argument n randam number qeneratar rfaHawed by hefunctmn name numbers rananna a pammrareetnaunan ra generate a errerent er ranean numbers qa ta tne rears menu chck Wameneet owns and enanae tne eeee vane an tne amen Vanaaree ah er tne eeee n W narteneet r e e r eeeenptran artne avanane and a gamma eretneunan nn he anen erneaetanetnhutmn Detnaunan generates rartne aeta rauanan 139 51 52 e e Xs1 10521 1151 F52 rne rananna aeeaeratee Wen tne aeta Detnaunan tax 51 52 wrtn tne arguments x e a ne er ar vectar ar rea va ues between an and 1 Nate Ta aHaw mteqratmn and atner averatmns averthxs argument va ues autsme artne stated range are aHawed but they Draduce a u resu t 51 and 52 e 725 shave Parameters qreatertnan u e a rea Wuhath between an and 1 e n mteqer qreatertnan u 3e The aeta Densrty Funmandhetagtlt5152r21urn5 tne Wabath densrty rar va ue x 51 e m 52 e A dbetam 651521845 A dbe axs152 db2a m4 ooo une ammw unemmA dbeta x 5 7 2 unemmm A h area underthe mman curve up m the va ue x M pbea 51 52 n 169 J dbetax5152 dx n 169 n h m Wabath v SDemfvthe Wabath v and mm returnsthe carresvanqu x qbe am 15951 52 M n m NSamp mnnn S vbetaWSamp e 51 52 Em the data faruse m the hxstaqram functmn EDm n 2 dbetaxs152 mes e 25 x u mes Number w hm Venar w hms am we ED e Mslugvam Use the WWW functer 50m an gt Sca e data based an number NSW E h afsamp esand hm me 4 2 The Elnnmlal DIstthutmn stmhutmn qeneratars farthe Bmarma Equatmn n k neb m n 1 n The hmarma arsenbunbn functmns abmbmw n a Returns the Wabath bensrcy farva ue k bbmbmw n a Remms the cumu atwe Wabath arsenbunbn rbr va ue k qubmw n a Re msthe mverse cumu atwe Wabath msmhutmn rbrbrbbebmcy v n thh the arguments k and n e mteqers wnere U 5 k 5 quot TD eubw mteqratmn and Ether averatmns averkr va ues auts de bnne stated range are aHawed buuney Waduce a u resuc e e a rea number between D and 1 b e a rea number between D and 1 m e an mteqer qreaterthan u bensrcy funman rs Walled he aw as aernbum He s br an eXDeHment whmh has exa vtwa bbssrbxe Dutcames sueeess and renure Vau canductthe exvemment Wabath br sueeess m each trwa The number abmbmw n n represents the Wabath bf exact v k suebesses nut br n Ma s n27 Ebmurmkm 3 The Band msmbuubn orserrbuubn qeneratars farthe Cauchv Equatmn The Cauchv msmhutmn functmns x w 5 Returns Wabath densxtv farva ue x peauehm 5 Returns the summath Wabath arsennuuan far va ue x uchv v w 5 Returns the m r52 summatva Wabath msmhutmn rar vmhahmtv v r Wren the arguments x e a ne er ar vectw af rea va ues 5 e a rea seaxe Parameter qreaterthan u p r a rea Wuhath between an and 1 m e an mags qreaterthan u Parameter Examme af a Cauchv arsennuuan n17 7 acauchvxxs nus e The n squared Distribution stmhutmn qeneratars farthe Ovrsquared Euuatmn The Cmrsquared msmhutmn funmans aemmx a Returns the Wabath aemwar va ue x nemsw a Returns the summath Wabath msmhutmn far v a ue x nemsqw a Returns the mverse summath Wabath msmhutmn rar Wabath v r Wren the arguments x e a sca ar a ctar ar rea va ues qreaterthan u Nate ra aHaw nteqratmn ne aperatmns averthxs argument va ues autsxde anne stated range are aHawed but they Draduce a u resu t meantta hefurmu ated far nteqer degrees affreedarnMath1ad aHaws rea va ues p n a rea vmhahmtv between u ang 1 m e an nteqer qreaterthan u n n 1 a mean gr u ang a standard dewatmn gr 1 Examme ar a cnnsguaneg ggennunan gemswmn 1 e 7 5 The Exvnnentlal DIstthutmn sttrxhutmn qeneratars farthe EXDanentxa Equatmn v expm x rne EXDanentxa dxstrxhutmn functmns dexvgtlt r Returns the Drahahmtv densth far va ue x DexDgtlt r Returns the curnu atwe Drahahmtv dxstrxhutmn far va ue x exDWr r Returns the nverse curnu atwe D ah htv dxstnhutmn far Drahahmtv D rexvm r Returns a vectar af rn randarn nurnhers newquot the expanentxa dxstrxhutmn mn the arguments x e a sca ar ar vectar ar rea va u25 qreaterthan ar equa ta u Ta a aHaw nQEqratmn ang v n 39 Draduce a u resu t r e a rea rate wasterth an n p n a rea vmhahmtv between u ang 1 m e an nteqer qreaterthan u expanerma functmn a the mm af u events 7 far examme thehme between fa ures afsame mechamca apparatus arthe wamnq we m a queue Examme af an EXDanerma stmhutmn dexmx 7 s The r Dlslnhutmn stmhutmn qeneratars farthe reo smauman Equatmn mnsm d2n5u2d1d2j rn5d1rn5d2 dam Xnsmm nsmez The r msmhutmn functmns m d1d2Retum the Wabath densxtvfar va ue x pm d1d2Retum the summath Wabath msmhutmn far va ue x UHFd1d2Retumsthemver52 e m x m m ahxhtv msmhutmnfarvrahahmtv v rFmd1d2Retum a vectar af m randam numbers havmq the r msmhutmn Wm the arguments x e a ne er ar vectw af rea va ues qreaterthan ar equax a u Ta aHaw mteqratmn and v h 39 Draduce a u resu t a and a2 7 rea a q wffreedam qreaterthan u v a 1 a rea Wuhath between an an an mteqer qreaterthan u the undenqu msmhutmn m nNovn testmq Examme m an r stmhutmn n5 7 grams 7 The Gamma Dlslnhutmn D smhutmn qeneratarsfarthe eamma Equatmn The eamma dxstmhutmn functmns dqammax 5 Returns vmhahmtv gem m va ue x anmmax 5 Returns mmu atwe Wabath msmhutmn m va ue x ggammav 5 Returns the mverse umu atw aha my dxstmhutmn farvrahahmtv v rqammam 5 Returns a vemaraf m randam numbers hawquot the gamma msmhutmn the the men the argumem5 x e a ne er a em af rea va ues qreaterthan u Ta aHaw mteqratmn and ather averatmns averthxs argument va ues autsxde afthe stated range are aHawed but they Draduce a u resu t 5 e a rea 5 ave Darameterqreaterthan u p 5 a rea Wuhath between u and 1 m e an ntetzer qreaterthan u The eamma dxstmhutmn 5 a mare qenera case afthe expanerma gemguan The gamma Examme wf a eamma sttmhutmn ggammawmz 7 a The Eenmetrm Dlslnhutmn stmhutmns assamated m the Geametm Equatmn k n l 7 n The Geametncdxsmhutmn functmns aqeamw a Rewms me whammy densxtv m va ue k e e 5 the cumu atwe Wabath msmhum m va ue k uqeamw q ReQurns the mverse cumu atwe Wabath msmhutmn farvmhahmtv v reemm a Returns a venar afm randam numbers hawquot the qeametncdxsmhutmn Wm the arguments k e a ne er ar vectar af rea va ues qreaterthan u Ta aHaw mteqratmn and ather V u va ues are aHawed but they waduee a u 725ml q r a rea Wuhath between an and 1 p r a rea Wuhath between an m e an mteqer qreaterlhan u w mdevendent aemaum He s Wm Sucuess Wabath a Exemme m a Geametru stmhutmn me EEEUMKIIZS cm H e m m m M m r 2 4 5 2 1 k 9 The HYDemenmetrK Distribution stmhutmns re ated m the HVDerqeametru Equemn The Hypergeometric distribution functions phypergeomm a b n Returns the cumulative probability distribution qhypergeomp a b n Returns the inverse cumulative probability distribution for probability p distribution with the arguments dhypergeomm a b n Returns the probability density for the hypergeometric distribution rhypergeomm a b n Returns a vector of m random numbers having the hypergeometric I mabandnintegerswhere 05 m5 a sn39 mSlJand Dsnsabaandbarethe quanities of the types of objects in a sample n is the number of samples chosen and p is the probability that exactly m of the n objects are type a objects To allow integration and other operations over these arguments values outside of the stated range are allowed but they produce a 0 result I p is a real probability between 0 and l The hypergeometric distribution is a finite discrete distributionThe hypergeometric distribution arises when sampling is taking place from a population with two types of objects a and b The sampling is done without replacement after one object is drawn it is not returned The hypergeometric distribution gives the probabilities of the different amounts of object a drawn Example of a Hypergeometric Distribution mU11IZI Ehypergaomm1015D2 il l il l U 2 4 6 8 1 U 10 The Logistic Distribution Distributions associated with the Logistic Equation x exp S x 1 2 s 1 exp S The Logistic distribution functions I dlogisx l s Returns the probability density for value x magma 5 Returns the summath Wabath msmhutmn m va ue x mama 5 Returns the m arse mmu atwe Wabath msmhutmn far Wabath v Mew 5 Returns a vectar af m randam numbers havmq the meme msmhutmn Wm the arguments x e a ne er ar vectw af rea va ues Parameter qreeterthan u v r a rea Wuhath between an and 1 m e an mteqer qreaterthan u Examme af a Lemme stmhutmn n1 mumsx53 5 n The annnrmal msmhmmn panama aeeaaaea me he Laanama saaanan 1 ex 2 7 w W e m m a x 2a The Laanama mam anamm densxtvfar va ue x D rmrmbm u 5 Returns summath Wabath msmhutmn m va ue x rmrmWy u 5 Returns m mverse summath Wabath msmhutmn far Wabath v ammo u 5 Returns the W the MM the arguments x e a ne er ar vectw af rea va ues qreaterthan ar equax a u Ta aHaw mteqratmn and V e they Waduce a u resu t u e a rea mean ame natura Dqs af swatmn ame n v r a rea Wuhath between an and m e an mteqer qreaterthan u atura Dqs af x qreaeennan u 1 af wages Damm ar v m the study gf Dam2 2 mncentratmns Exemme af 5 L qnarma sttnhutmn g1 54 gxnuvmxugn2 12111 Negame Elnnmlal DIslthutmn stmhutmns assamated mn the Negeme Bmarma Equatmn af sue n nekex quot k k JR n The Negatwe Bmarma dxstnhutmn functmns n n g summatva Wabath msmhutmn m va ue k h v v mhmamm n v Remms a vectar af m randam numbers havmq the negeme hmarma dxstnhutmn mn the arguments n and k e mteqers mn n s u and k 3 U Ta aHaw mteqratmn and ather averatmns aver these arguments va ues autsxde gnne stated range are aHawed m hay Druduce a u resu t g e a rea whammy mil yen nee prahahmtv between D eng 1 en neegen qreaterthan e as The negeme hmarma dxstnhutmn 5 an m mte dxscrete msmhutmn Exemme m a Negeme Bmarma sttnhutmn dnbmumk5 3m as n n m 2U 3U 40 n k 13 The Nnrmal memhumn momma assamated m the Narma swam X 2 2 u The Narma mama functmns dnarmbm U y H Retumsthe Drahahmtv densxtv far va ue x Dnarmgtlty U y 5 Retumsthe cum Mat v2 Wabath ammuuan farva ue cnarmgtlt Returns the cumu atwe vmhahmtv msmhutmn Wm mean u and vananue 1 mm u 5 Returns the mverse summath Drahahmtv msmhutmn far Wabath v marmUm u 5 Returns a vectar af m randam numbers hawquot the narma msmhutmn Wm the arguments x e a ne er ar vectw af rea va ues u e a rea mean a e a rea standard dewatmn v r a rea Wuhath hetwe m e ranmeq greaterthan an en u and 1 er qreaterthan u The Narma ar Gaussxam msmhutmn 5 svmmetru ahaut ms rmdd e va ue at x e m and has mm can m exther dxrectmn wmeh avvmach m nevertaum the Xe x Examme m a Narma sttnhutmn anavrnxuan n5 7 u The Pnlssnn Dlslnhutmn stmhutmns re atmq a the pawn Equatmn The pawn msmhutmn functmns dvmsk A Returns the Wabath densxtvfar va ue p ppaw A Retumsthe cumu atwe pnaaaamy awaauan farva ue p upaw A Returns mverse summath pnaaaamy msmhutmn far Wuhath p rDm5m A Returns a vectar af m randam numbers havmq the pawn msmhutmn mn the arguments p r a p argument va ues autsxde anne stated range are aHawed but they Dmduce a u resu t A r a rea mean qreaterthan u p r a rea Wuhath between an ana 1 m 7 an mteqer qreaterthan u The pawn msmhutmn 5 anather examme af an m mte dxscrete msmhutmn Examme af a pawn stmhutmn n2 Epm k e n1 15111 smdem Distributlnn stmhutmns assauated Wm the Student s T Euuatmn r zjensmex 2 x The Student 2 msmnucmn mnctmns am a Returns the Wabath aenm farva ue x pm a Returns the summath vmhahmtv msmhutmn m va ue x new a Returnsthe mverse cum 2 e Drahahmtvdxsmhutmnfarvrahahmtvv mm a Returns a vectw w m randam numbers hawquot the Student s 2 mman Wm the arguments x e a ne er ar vectw w rea va ues a e an mteqer degree affreedam qreaterthan u Nate that mm the msmnucmn lager va ues v r a rea Wuhath between an and 1 m e an mteqer qreaterlhan u Exemme w a Student Dmmumn mwmr 16111 Uniform mstnhutmn Drstrmutmns assamated thh the Umfarm Equatmn a ue x mm a h Returns the summatva Wabath msmhutmn far va ue x r R ve Wabath metnnutmn rar Wabath v x m msmhuted r dam number between an and x rum mr a h Returns a vectar af m randam numbers hawquot the umfarm msmhutmn thh the arguments x e a ne er ar vectar af rea va ues between a and h nc uswe Ta aHaw mteqretmn and r r they praduee a u resu t m e an mteqer qreetert an n mare hkehhaad af appeermq than anv ather pamt Exemme er a Umfmm orstrmutmn dunWXabU5 7 n The Welhull Distribution stmhutmns assamated Wren the Wernuu Euuatmn 5 1 5 s x exp x The Wernuu msmhutmn functmns ewemumx 5 Returns the Wabath eenm farva ue x pwerhum 5 Returns the summath Wuhath msmhutmn far va ue x ewemumv 5 Returns the mverse summath p ah my ersermumn far Wabath v rwemuum 5 Returns a vectw af m randam numbers havmq the Wemuu mscrmucmn Wren the arguments x r a ne er ar vectw af rea va ues qreaterthan ar equa m u Ta aHaw mceeramn and r r they praeuee a u rem 5 r a rea shave Parameter greaterthan u p r a rea Wuhath between u and 1 e u m 7 an m eeer qreaterthan the Wernuu dxstnhutmn amses r m radar anew Examme af 5 Wernuu stmhutmn dwemqu mS

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