Modifed after PowerPoint by Fauziah Binti AzizThe Binomial, Poisson, and Normal DistributionsProbability distributionsOWe use probability distributions because they work they ft lots of data in real worldHt (cm) 1996100806040200Std. Dev = 14.76Mean = 35.3N = 713.00Height (cm) of Hypericum cumulicola at Archbold Biological StationRando !ariableO"he atheatical rule #or function$ that assi%ns a %i!en nuerical !alue to each possible outcoe of an e&perient in the saple space of interest'Rando !ariables O(iscrete rando !ariablesO)ontinuous rando !ariables"he Binoial (istributionBernoulli Rando *ariablesO+a%ine a siple trial with only two possible outcoesOSuccess (p)OFailure (q)OExamplesO"oss of a coin #heads or tails$O,e& of a newborn #ale or feale$O,ur!i!al of an or%anis in a re%ion #li!e or die$Jacob Bernoulli (1654-1705)"he Binoial (istributionO!er!iewO,uppose that the probability of success is pOWhat is the probability of failure-Oq . / pOExamplesO"oss of a coin #S . head$0 p . 1'2 q . 1'2ORoll of a die #S . /$0 p . 1'/334 q . 1'5666OFertility of a chicken e%% #S . fertile$0 p . 1'5 q . 1'7"he Binoial (istributionO!er!iewO+a%ine that a trial is repeated n tiesOExamplesOA coin is tossed 2 tiesOA die is rolled 72 tiesO21 chicken e%%s are e&ainedOAssue p reains constant fro trial to trial and that the trials are statistically independent of each other"he Binoial (istributionO!er!iewOWhat is the probability of obtainin% x successes in n trials-OExampleOWhat is the probability of obtainin% 7 heads fro a coin that was tossed 2 ties-P#HHTTT$ . #/87$2. /867"he Binoial (istributionO!er!iewOBut there are ore possibilities099""" 9"9"" 9""9" 9"""9"99"" "9"9" "9""9""99" ""9"9"""99P#7 heads$ . /1 : /867 . /1867"he Binoial (istributionO!er!iewO+n %eneral; if trials result in a series of success and failures;FFSFFFFSFSFSSFFFFFSF"hen the probability of x successes in that order isP#x$ . q q p q . px qn x"he Binoial (istributionO!er!iewO9owe!er; if order is not iportant; thenwhere is the nuber of ways to obtain x successesin n trials; and i< . i #i /$ #i 7$ = 7 /nx(n x)p px x q qn x n xP P( (x x) ! ) !nx(n x)"he Binoial (istributionO!er!iew"he Poisson (istributionO!er!iewOWhen there is a lar%e nuber of trials; but a sall probability of success; binoial calculation becoes ipracticalO>&aple0 ?uber of deaths fro horse kicks in the Ary in di@erent yearsO"he ean nuber of successes fro n trials is . npO>&aple0 3A deaths in 71 years fro thousands of soldiersSimeon "# $oi%%on (17&1-1&40)"he ?oral (istributionO!er!iewO(isco!ered in /466 by de Moi!re as an appro&iation to the binoial distribution when the nuber of trails is lar%eO(eri!ed in /51B by CaussO+portance lies in the )entral Diit "heore; which states that the su of a lar%e nuber of independent rando !ariables #binoial; Poisson; etc'$ will appro&iate a noral distributionOExample0 9uan hei%ht is deterined by a lar%e nuber of factors; both %enetic and en!ironental; which are additi!e in their e@ects' "hus; it follows a noral distribution''arl (# )au%% (1777-1&55)Abraham de *oi+re (1667-1754)"he ?oral (istributionO!er!iewOA continuous rando !ariable is said to be norally distributed with ean and !ariance 7 if its probability density function isOf#x$ is not the sae as P#x$OP#x$ would be 1 for e!ery x because the noral distribution is continuousO9owe!er; P#x/ E X F x7$ . f#x$dxf (x)! 1,(x ),-,,ex x1 1x x, ,"he ?oral (istributionO!er!iew"he ?oral (istributionO!er!iew"he ?oral (istributionO!er!iew*ean change% .ariance change%"he ?oral (istributionDen%th of FishOA saple of rock cod in Monterey Bay su%%ests that the ean len%th of these fsh is . 61 in' and 7 . A in'OAssue that the len%th of rock cod is a noral rando !ariableO+f we catch one of these fsh in Monterey Bay;OWhat is the probability that it will be at least 6/ in' lon%-O"hat it will be no ore than 67 in' lon%-O"hat its len%th will be between 73 and 7B ilietres-"he ?oral (istributionDen%th of FishOWhat is the probability that it will be at least 6/ ' lon%-"he ?oral (istributionDen%th of FishOThat it will be no more than 32 in. long?"he ?oral (istributionDen%th of FishO"hat its len%th will be between 73 and 7B inches-,tandard ?oral (istributionOG.1 and H7./-6 -4 -2 0 2 4010002000300040005000Iseful properties of the noral distribution/' "he noral distribution has useful properties0O)an be added >#JKL$. >#J$K>#L$ and 7#JKL$. 7#J$K 7#L$O)an be transfored with shift and change of scale operations>&ercise /ocation of the mea%urement