Wednesday s lecture. Sums of normal random variables. Some examples, n=1. Central Limit Theorem
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1 Wedesday s lecture A liear combiatio of ormal radom variables is ormal Sums of ormal radom variables Let X ~ N(µ,σ 2 ) ad Y ~ N(ν,τ 2 ) be idepedet. Fact: Liear combiatios of ormal radom variables are ormal. To figure out the distributio of Z=aX+bY we eed oly determie the mea ad variace of Z. But E(Z) = Var(Z) = What chages if X ad Y are correlated? Cetral Limit Theorem The ormal distributio arises aturally wheever we loo at a sum of a large umber of iid radom variables Some eamples, =1 Ep(1) U(0,1) Theorem 5.3.B: Let X 1,,X be iid radom variables with epected value µ ad variace σ 2. For ay real umber lim X i µ i=1 P < z = σ 1 2π z e 2 /2 d = Φ(z) Geom(1/3) Po(1)
2 Some eamples, =15 Epoetial Uiform Legth of overtime i NHL playoff games I 24 years of NHL playoffs, 251 playoff games wet ito overtime Geometric Poisso We ca draw samples of size from this distributio ad loo at their averages Other sample averages, NHL overtime A biomial approimatio Whe is small it is easy to compute biomial probabilities eactly. But what about whe is large? De Moivre foud the ormal approimatio to biomial (what we ow ca use the CLT for) X = i=1 Y i E(X)= Var(X)= F(+1/2) F( 1/2) f()
3 p=1/3 The Poisso distributio = = Number of rare evets i fied time period ad/or area Biomial with large ad small p If X ~ Bi(, λ/) the X ~ Po(λ) i the sese that = =100 # lim% $ & ' ( # λ & % $ ( ' # 1 λ & % ( $ ' = λ! e λ Siméo Deis Poisso ( ) Productio cotrol Assembly-lie processes do ot yield costat output. There is always variatio i the quality of items produced. Cotrol charts are used to moitor whether quality variatio ca reasoably be assiged to chace or there are other sources of variatio to loo for. A mea chart for process cotrol loos at successive averages of quality measures (such as measures of size). Based o previous data, cotrol limits are determied. If the plot of successive averages fall outside the cotrol limits, the process is out of cotrol, ad correctios must be made. Productio cotrol, cot. By the cetral limit theorem, the averages are approimately ormally distributed, with µ = process mea ad σ = process stadard deviatio / 1/2, where is the umber of measures averaged. Thus we epect the process to eceed ± 3σ oly rarely. If we do ot ow the process stadard deviatio it must be estimated from the data, ad the umber 3 will be replaced by a slightly larger umber.
4 A cotrol chart What causes lac of cotrol? Chage i level: differet supplier differet operator Tred: machie movig out of aligmet deterioratio i quality of raw materials Chage i dispersio: differet supplier ot detectable by meas chart Food processig A ice cream compay produces 200 ouce ice cream cotaiers. The process is desiged to produce actual fill amouts betwee 196 ad 204 ouces. At 10- miute itervals, four cotaiers were tae from the productio ad the average weight computed. The overall mea of 24 such samples (96 cotaiers) was ouces, with a estimated stadard error for each average of four of Igorig the estimatio of the stadard error, the cotrol limits are 204 ± Chapter 7 Discrete distributios Geometric Negative biomial Poisso The Poisso process
5 Radom stoppig A etomologist is attemptig to capture a elusive butterfly. I a meadow there are 10% butterflies of the elusive id, ad 90% of a similar-looig commo id. I order to determie whether or ot a catch is the elusive oe, the etomologist eeds to ispect the legs. If they are smooth, the butterfly is of the elusive id, while if they are hairy it is of the commo id. How may butterflies will the etomologist have to catch i order to get a elusive oe? X = # trials eeded to catch a elusive butterfly P( X = ) = pq -1, =1,2,... Feedig behavior of birds The followig data were collected i a ecological study. It cosists of the umber of hops betwee flights for several birds of the same species. # of hops Observed Epected X has a geometric distributio A property of the geometric distributio The geometric distributio has o memory: P(X = +) pq = +-1 P(X > ) P(X = + X > ) = = pq -1 = P( X = ) i=+1 pq i-1 = pq+-1 pq 1-q More butterflies Suppose our etomologist is iterested i catchig r of the rare butterflies, ot just oe. Let X be the umber of butterflies of either id eeded to get r rare oes. X has a egative biomial distributio. We write X ~ NegBi(r,p). P( X = ) =
6 Isect couts From each of 6 apple trees i a orchard that had bee sprayed, 25 leaves were selected. O each leaf the umber of adult female red mites were couted. A Poisso distributio would apply if the ifestatio were radom, but a higher variability would be epected if the isects ted to be clustered. The egative biomial distributio ofte applies to clustered situatios. Number Obs Po NegBi (- ) = Geeral biomial coefficiets (--+1)(--+2) (-) = (-1) 1 (+-1) 1 Here eed ot be a iteger. Now write ( ) =! P( X = r + ) =!(-)! = (-+1)(-+2) 1 r+-1 ( ) p r q = ( ) -r p r (-q) =(-1) ( ) +-1 Is there a r'th success? It may ot be obvious that there has to be a r'th success evetually, i.e., that the egative biomial pdf sums to 1. We eed to compute =0 -r p r (-q) P( X = r + ) = =0 We eed to use Newto's biomial formula Hece so (1+t) a = 1 + a 1 t + a 2 t2 + =0 -r (-q) = (1-q ) -r = p -r P( X = r + ) =0 = p r p -r = 1 Natioal Hocey League The followig data are o the umber of goals scored i the Natioal Hocey League durig the seaso. There were a total of 420 games. Goals Freq The average umber of goals per game was 3.0 = rq/p while the variace of that umber was 3.4 = rq/p 2 This correspods to p = 0.85, r = 16.5.
7 Radioactive disitegratio Leuemia clusters A famous cluster of leuemia cases amog childre occurred i Niles, IL, Amog the 7076 childre uder 15 years of age, 8 developed leuemia. I all of the couty, there were 286 cases amog 1,152,695 childre at ris. The couty-wide icidece rate for the 51/3 years uder cosideratio was I 1910 Rutherford ad Geiger studied alpha particles emitted by poloium ad producig scitillatios at a scree, the recordig each such flash o a chroograph tape scitillatios couted over 5 days ,000 = 24.8 cases per 100,000 1,152,695 Is the Niles icidece rate uusual? = p = Epected umber of successes = P ( X 8 ) = Has Geiger Erest Rutherford Horse-ic fatalities Outbreas of wars Prussia cavalry late 19th cetury: 10 corps, 20 years Statistics o cavalry soldiers illed by horse-ics Number of Observed deaths umber of corps-years Total umber of deaths = 122. Death rate 122 / 200 = 0.61 deaths per corps-year. What idepedece assumptios do we mae? I the 432 years from 1500 to 1931, there were 299 wars recorded i the world. (Major wars were couted as several sub-wars). Number of wars Observed Epected startig frequecy frequecy Is there a costat huma hostility level?
8 A Poisso process path Poisso process Cout the umber X(t) of evets i (0,t]. Assume X(t) ~ Po(λt) Evets i disjoit itervals are idepedet Y = time util first evet. P(Y > t ) = P(X(t) = 0) = X(t) Li Evets i a Poisso process are upredictable, i the sese that if you ow how may already happeed it does ot help to predict the et. T 1 T 2 T 3 T 4 t T 5 X(t) is a coutig process Y = T -T -1 has desity λe -λ, > 0
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