Random numbers and random events

Transcription

1 4// Radom walks, etc. Radom evets Radom umbers ad radom evets Mathcad has a bult fucto rd(x) that selects a umber radomly o the terval to x. If x =, ths meas that % of the umbers wll be less tha. ad % wll be betwee. ad.. We ca check ths by makg a lst of of umbers created by rd() ad the plottg the dstrbuto. The umber of radom umbers wll be: rd( ) Ths programmg loop wll fd values ad put them the vector whch wll have elemets. We wll ot use the frst elemet. Now let's plot the dstrbuto as a hstogram. Frst we dvde the doma (,) to tervals. The vector cotag the tervals wll have values:,..... tervals. H hst( tervals) The hst fucto couts the umber of etres each terval ad returs a vector wth oe fewer elemets tha terval values. Ths vector, H, cotas the data the hstogram plot below. H 5 Dstrbuto of radom umbers Ths s a -d plot wth "soldbar" selected as the le type usg the graph dalog box. The horzotal axs has.5 added to so that the bars are plotted the mddle of the terval, ot at the begg of the terval. Ths meas that the bar values to. s cetered o Ispecto shows that deed about % of the radom umbers fall each terval.

2 4// Radom walks, etc. Radom evets We expect / = umbers couted each b. What s the stadard devato of the umber the bs (call t Sdev)? Sdev H Sdev was last defed as gog from to so there are bs. The varato from oe sample to the ext s about +7. Note that the stadard devato s defed so that the dvso s by oe less tha the umber of bs. Try t: Chage from to, ad watch what happes to the stadard devato. Is t a larger or smaller percetage of the H vaues? Radoactve decay H Suppose a radosotope geerates, o average, oe cout about every secods. That meas there s a % probablty each secod of there beg a decay. How ca we model that? Let's select a radom umber usg rd(). The there s a % chace of the umber beg below.. We wll say that there has bee a decay f rd() s less tha. ad o decay f t s ot. We wll keep a record of decays the followg way: We wll set the elemets of a vector tally to zeroes ad the chage the zero to f there s a decay. For ths example, "" meas true ad "" meas false (o decay). If there s a decay the frst secod, s, otherwse t s zero. The couter below couts the secods. f ( rd( ). ) Now let's plot k 6 The -d plot below has "stem" selected as the trace type whch puts crcles o stalks. The vertcal scale s set to. The stems are radomly spaced. The clusterg at certa places ad gaps at other places are radom! We plot oly the frst 6 secods cotag about 6 radomly spaced decays. Tmes of decays.5 k k The pots dcate the tmes whe the "clcks" were heard o the geger couter.

3 4// Radom walks, etc. Radom evets We ca fd the umber of decays by smply summg the "" values the vector : k 6 k k Ideed there are about decays tme tervals f the probablty s % each terval. Try t: If you select Calculate Worksheet m the Math meu, the calculato wll be doe aga wth a ew set of radom umbers. Does ths chage by much the umber of decays? Selectg radom tme tervals Suppose you wat to fd the tme terval betwee oe decay ad the ext. How would that be doe? Suppose the decay tme s. If we select a radom umber o the doma (,), the fucto - l[rd()] coverts the radom umber to a tme terval. Sce the log of rd() wll be less tha oe we put a mus sg frot of the logarthm to create a postve umber from the egatve logarthm. If rd() s. the the logarthm s -6. whch becomes +6.. Thus log tme tervals (t >> ) are very ulkely because they requre a very small value rd(). Try t: Show that the probablty of 6. decay tmes passg wthout a decay s.. O the other had, f rd() s., the tme terval geerated s +.. Thus tervals betwee ad. are geerated wth % probablty. So ths fucto seems to be what we eed. Let's check that. τ Ths s the decay tme correspodg to. probablty per secod. Number of decays. τ l( rd( ) ) Italze the last to talze all values ad to reserve memory space all of. ow cotas the tme tervals betwee decays. We have to sum the tervals to get the tme elapsed sce t =. k 6 k k m m cotas the tmes of the decays. Itervals betwee decays:

4 4// Radom walks, etc. Radom evets 4 Radom tmes from summg -l[rd()] k tervals As our prevous graph, some of the tervals are qute log ad others are short. Let's plot a hstogram of the legths of the tervals. The wdths of the bars the plot wll be sec. H hst( tervals) 5 H exp τ 5 The tervals have a dstrbuto whch s expoetal as exp(-t/). Thus our fucto - l[rd()] gves the correct "mappg" of radom umbers to decay tme tervals.

5 4// Radom walks, etc. Radom evets 5 Smulatg the decay of a populato If the probablty s. per secod, the probablty that a atom has ot decayed s gve by exp(-t/), show at rght. We wll follow atoms secods. The decay tme = secods. Each secod, the probablty of decay s.. max jmax j couts secods, so the tme terval s Δt t e t. t P P max j jmax P j max j jmax Δt P f rd( ) j τ P j Italze P values to usg ested loops. Each row of P represets a ucleus. The row wll have value f the ucleus has ot decayed ad value f t has decayed. The "f" statemet sets P,j = f there s a decay, otherwse P,j stays at the prevous value of or. The decay probablty s.. j s the tme step umber ad s the partcle umber. I the matrx P, each partcle s oe row. If the partcle decays o the th tme step, the value chages from to ad stays zero. Tme creases to the rght. P The frst colum of P s all oes because oe of the partcles have decayed at t =.

6 4// Radom walks, etc. Radom evets 6 We ca sum the colums to see how may partcles are stll left after each tme terval. There are max+ partcles to start: jmax j jmax After j tme tervals, the umber remag s We expect the aswer to be expoetal T Ftheory maxe T τ max P max F j P j Number remag: F Expoetal decay 48 7 F Ftheory T Selectg radomly from a populato A group of gas atoms wth a Maxwella velocty dstrbuto three dmesos has a speed dstrbuto gve by Frst defe a thermal velocty: v t The Maxwella: g( v) 4πv πv t e v v t A set of values plottg: v.

7 4// Radom walks, etc. Radom evets 7 How would you select speeds radomly from ths dstrbuto? Place a pot X,Y radomly o the graph ad accept t f t the "Y" value uder the curve ad reject t f t s above the curve. At each value of v, the heght of the dstrbuto s proportoal to the probablty. So we choose a guess v radomly o the terval,. The the probablty of acceptg the guess s proportoal to heght of g(v). Ths s called the "rejecto method" because we wll throw away the guesses at pots above the curve. gv ( ) Dstrbuto of Maxwella speeds Select speeds from the Maxwella. v y rd( ) v rd( ) whle y g( v) v rd( ) y rd( ) v Italze ad reserve memory space. y s a radom y-axs value o the graph above. v s a radom guess v. The "whle" loop cotues to make guesses y ad v. The probablty that v s kept s determed by the heght of the curve. If the radom umber y s below the heght of the curve, the guess v s kept, otherwse t s rejected ad a ew value s tred. Let's make a hstogram of our radom guesses: Prepare a hstogram: tervals w. hst( tervals) V 8 w 6 gv 4 Maxwella speeds selected by rejecto I the chart dalog box, uder traces, clck o sold bar to make a hstogram plot. Use +.5 o the x axs to ceters the bars. 4.5 Try t: Is the radomly chose sample more lke the theory curve f s creased to,? to,? Note: Our guess y was o the terval, because the maxmum value of g(v) s a lttle less tha oe. If the maxmum of g(v) were hgher tha, we would have to crease the maxmum allowed value of the radom guess.