Stirling s Formula and DeMoivre-Laplace Central Limit Theorem

Transcription

1 Stirlig s Formula ad DeMoivre-Laplace Cetral Limit Theorem Márto Balázs ad Bálit Tóth October 6, 04 Uiversity of Bristol / Budapest Uiversity of Techology ad Ecoomics

2 Usig Stirlig s formula we prove oe of the most importat theorems i probability theory, the DeMoivre- Laplace Theorem. The statemet will be that uder the appropriate ad differet from the oe i the Poisso approximatio! scalig the Biomial distributio coverges to Normal. First we prove Stirlig s formula for approximatig factorials. This will be eeded to estimate the biomial coefficiet i the Biomial distributio. Theorem Stirlig s-formula For large iteger s we have! e π, to be more precise > 0 <! e π < e For large s, the error term that bouds the fractio is approximately +. Proof. The value of π will ot result from this proof yet. We will use the DeMoivre-Laplace Theorem to see this. Takig the logarithm of!, l! lk. We start with some ituitio that explais why thigs are doe later the way as show. Comparig areas o the pictures, we have 0 k lxdx lk k + lxdx :. l l7 l6 l5 l4 l lx l l7 l6 l5 l4 l lx l l x x Computig the itegrals, l l! +l+. Our approximatig formula will be betwee these two bouds. We pick + /l, ad look at the differece [ d l! + ] l. We show that d is mootoe decreasig, thus coverges to a limit d, for every, we also have d < d < d+. Oce we have these properties, the statemet follows: e d < e d! +/ e < e d e, that is, <! e π < e with some ukow costat π. To see the two properties, we look at the decremet of d : d d + l+ + l+ t + + l + l l +t t t[ l+t l t ], + l+

3 where we itorduced the ew variable t /[+]. Itegratig the sum formula t for the geometric series, ad lookig at the substitutio at t 0 we easily derive the Taylor series of l t: t k l t k, from which l+t ktk k also follows. Usig this, d d + t l k k0 t k k tk k t k l0 k t l+ l+ t l t l+ l+, which shows that d is decreasig, therefore it has a limit d. We proceed with the sum o the right had-side: t l d d + < t t + + +, from which d is icreasig. Thus d d lim d d m lim m m d d m + m + m < lim m m. Let ow X Biom, p, p fixed, ad take to ifiity. This is very differet from the Poisso approximatio, where p wet to zero like the reciprocal of. Below we use q p. The stadardised versio X EX DX X p pq of the Biomial variable has zero mea ad stadard deviatio oe. The statemet of the DeMoivre-Laplace Theorem will be that as, this stadardized Biomial distributio coverges to Stadard Normal. However, as the Biomial is discrete ad the Normal is cotiuous, it is ot eve clear at first readig how to formulate such a statemet precisely. Cosider the probability mass fuctio of the Biomial variable X. s X is iteger-valued, it is atural to associate the iteger k with the real iterval [k /, k +/: p X k PX k} P k X < k + } k / p P pq X p pq < k +/ p pq }. The legth / pq of this rescaled iterval goes to zero. By the ituitive meaig of the probability desity fuctio, we are ow able to formulate what is meat by the covergece of the rescaled Biomial to the Normal distributio: for large we expect k / p p X k P pq X p < k +/ p } k p ϕ pq pq pq pq, where ϕ is the Stadard Normal desity. This is the essece of the statemet of the DeMoivre-Laplace Theorem: Theorem DeMoivre-Laplace Let X Biom, p, where p is fixed. Let be odecreasig such that lim / /6 0. The p X k pq k p < O ϕ k p pq that is, the left had-side stays bouded i after dividig it by /. Proof. It will be a cosequece of this theorem that the costat π equals π, we do ot eed to assume this. pproximatig the Biomial mass fuctio via Stirlig s formula, π +O πk π k p X k! pk q k k! k! πpq πpq p k q k k k k k k q k k p q k p k k k q / k + k p p } } I k p / +O k p q / / + k p p } } II +O.

4 The advatage here is that accordig to the coditio i the, we have [k p]/ < / which goes to zero. Therefore takig logarithm of the term I ad expadig it to secod order we have li k l k p k l + k p q p k k p q kp kq pq k k p p k k p q k p+ kp +kq p q k p +O k p + k p }}}} p k p+q k p+ p kp q p+q pq p q k p +O pq +p kp q p q k p +O pq +pp q kp q p q k p +O k p p qk p pq p q +O k p +O, pq sice the secod term is also O /. Take the expoetial agai: I e k p pq +O. +O I the term II we oly look at the costat part of the series expasio, with first order error boud: Combiig the details, p X k II +O. e k p pq +O +O +O πpq e k p pq +O +O +O πpq e k p pq +O πpq where we picked the largest of the error bouds i the last step. I the DeMoivre-Laplace Theorem mass fuctios ad desitiy fuctios are featured, thus we have a local limit theorem. Notice that fie asymptotics ad error bouds are also part of the statemet. The versio comparig distributio fuctios is the global limit theorem: Theorem Global form of the DeMoivre-Laplace Theorem Let p be fixed, ad X Biom, p. The for all a < b fixed reals, we have lim P a X p } < b Φb Φa. pq Proof. By the triagle iequality, P a X p < b } Φb Φa b pq +p b p X k ϕx dx pq k a pq+p b pq +p k a pq+p + b pq +p k a pq+p a p X k k p ϕ pq pq k p ϕ b pq pq a ϕx dx.

5 The secod term is the differece betwee the itegral of the fuctio ϕ ad the Riema sum of this itegral, therefore this term goes to zero. For the first term, let c > b pq, a pq. Rewritig the DeMoivre-Laplace Theorem, k p <c p X k k p ϕ pq pq k p <c p Xk pq ϕ pq k p O O πpq k p ϕ pq pq for all k s i. The sum has oly O may summads, thus the sum of the error bouds is still i the order of O/. The global theorem is ow proved for the desity ϕx / π e x /. However, it the follows that this fuctio ϕ is a proper desity fuctio, from which we also coclude that π π. We remark here that the global theorem is a special case of the so-called Cetral Limit Theorem. Example 4 The ideal size of a course is 50 studets. O average 0% of those accepted will eroll, therefore the orgaisers accept 450 studets. What is the probability that more tha 50 studets eroll? ssumig idepedece, the umber of those who eroll is X Biom450, 0.. Usig the global theorem, p 5, pq 9.7 X 5 PX > 50} PX > 50.5} P > } Φ Φ Example 5 Flippig a fair coi 40 times, what is the probability that exactly 0 Heads result? The aswer to this questio is of course 40 0 / approxmative aswer ca be give usig the DeMoivre-Laplace Theorem: p0 pq ϕ 0 p pq 0π e We ca also use the global versio of the theorem, X is the umber of Heads that result: 9.5 p PX 0} P9.5 X < 0.5} P X p < 0.5 p } pq pq pq P 0.5 X 0 < 0.5 } Φ Φ Φ Below we plot the graphs coected to this example. The solid lie represets the stadard ormal desity fuctio the deomiator i the DeMoivre-Laplace Theorem. Dots plot pq 0 times the probability that the Biomial radom variable takes o value pq x+p 0 x+0 whe this is a iteger this is the umerator i the DeMoivre Laplace Theorem. Oe ca hardly spot ay differece. ϕx x