Further Topics on Random Variables: Transforms (Moment Generating Functions)
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1 Furthr Toic on Random Variabl: Tranform (omnt Gnrating Function) Brlin Chn Dartmnt of Comutr Scinc & Information Enginring National Taiwan Normal Univrity Rfrnc: - D. P. Brtka J. N. Titikli Introduction to Probability Sction 4.
2 Aim of Thi Chatr Introduc mthod that ar uful in Daling with th um of indndnt random variabl including th ca whr th numbr of random variabl i itlf random Addring roblm of timation or rdiction of an unknown random variabl on th bai of obrvd valu of othr random variabl Probability-Brlin Chn
3 Tranform Alo calld momnt gnrating function of random variabl Th tranform of th ditribution of a random variabl i a function of a fr aramtr dfind by If i dicrt If i continuou () [ ] E f d Probability-Brlin Chn
4 Illutrativ Eaml (/5) Eaml 4.. Lt / / / if if if 5. Notic that : E[ ] () [ ] E 5 Probability-Brlin Chn 4
5 Probability-Brlin Chn 5 Illutrativ Eaml (/5) Eaml 4.. Th Tranform of a Poion Random Variabl. Conidr a Poion random variabl with aramtr : K! ()!! claurin ri Lt!! a a a a a a a a L Q
6 Illutrativ Eaml (/5) Eaml 4.. Th Tranform of an Eonntial Random Variabl. Lt b an onntial random variabl with aramtr : f ( ) ( ) d d Notic ( if < ) that : () can b calculatd only whn < Probability-Brlin Chn
7 Illutrativ Eaml (4/5) Eaml 4.4. Th Tranform of a Linar Function of a Random Variabl. Lt b th tranform aociatd with a random variabl. Conidr a nw random variabl a b. W thn hav [ ] b [ a] b E ( a) () ( a b E ) For aml if i onntial with aramtr and thn () () Probability-Brlin Chn 7
8 Illutrativ Eaml (5/5) Eaml 4.5. Th Tranform of a Normal Random Variabl. Lt b normal with man μ and varianc σ. W firt calculat th tranform of a tandard normal random variabl f ( y) () π / / / y y / y π π π / dy [( y / ) y ( / ) ] ( y) / dy dy Sinc w alo know that w can hav σ μ μ () ( σ ) μ μ σ / ( σ / ) -μ σ Probability-Brlin Chn 8
9 From Tranform to omnt (/) Givn a random variabl w hav () [ ] E f ( )d (If i continuou) Or () [ ] E Whn taking th drivativ of th abov function with rct to (for aml th continuou ca) d d If w valuat it at w can furthr hav d d () d f () d d (If i dicrt) f d th firt momnt of f d f d E[ ] Probability-Brlin Chn 9
10 From Tranform to omnt (/) or gnrally th diffrntiation of n tim with rct to will yild d n d n () n f n n d f d E[ ] th n-th momnt of Probability-Brlin Chn
11 Probability-Brlin Chn Illutrativ Eaml (/) Eaml 4.a. Givn a random variabl with PF: 5. if / if / if / () [ ] 5 E () ] [ 5 d d E () ] [ 5 d d E
12 Probability-Brlin Chn Illutrativ Eaml (/) Eaml 4.b. Givn an onntial random variabl with PF:. f d d < if [ ] d d E [ ] d d E
13 Two Prorti of Tranform For any random variabl w hav [ ] E E[] If random variabl only tak nonngativ intgr valu ( L ) P( ) lim lim k () lim P( k ) P( ) k Probability-Brlin Chn
14 Invrion of Tranform Invrion Prorty Th tranform aociatd with a random variabl uniquly dtrmin th robability law of auming that i finit for all in an intrval [ a a] a () Th dtrmination of th robability law of a random variabl > Th PDF and CDF [ ] In articular if for all in a a thn th random variabl and hav th am robability law Probability-Brlin Chn 4
15 Probability-Brlin Chn 5 Illutrativ Eaml (/) Eaml 4.7. W ar told that th tranform aociatd with a random variabl i () () hav w will i dicrt) (if w comar th formula If P P P P
16 Probability-Brlin Chn Illutrativ Eaml (/) Eaml 4.8. Th Tranform of a Gomtric Random Variabl. W ar told that th tranform aociatd with random variabl i of th form Whr () i a gomtricrandom variabl i a dicrt random variabl with PDF - It can b infrd that thn rd a i - Bad on th rorty that -. w can t If < < K K K α α α α α () < [ ] d d d d E
17 itur of Ditribution of Random Variabl (/) Lt K n b continuou random variabl with PDF f f n and lt b a random variabl n which i qual to with robability ( ). Thn K i i f i i f L f n n and L n n Probability-Brlin Chn 7
18 itur of Ditribution of Random Variabl (/) itur of Gauian Ditribution or coml ditribution with multil local maima can b aroimatd by Gauian (a unimodal ditribution) mitur f n ( ) y N y; μ σ i i i i i n i i Gauian mitur with nough mitur comonnt can aroimat any ditribution Probability-Brlin Chn 8
19 An Illutrativ Eaml (/) Eaml 4.9. Th Tranform of a itur of Two Ditribution. Th nighborhood bank ha thr tllr two of thm fat on low. Th tim to ait a cutomr i onntially ditributd with aramtr at th fat tllr and 4 at th low tllr. Jan ntr th bank and choo a tllr at random ach on with robability /. Find th PDF of th tim it tak to ait Jan and th aociatd tranform Probability-Brlin Chn 9
20 Probability-Brlin Chn An Illutrativ Eaml (/) Th rvic tim of ach tllr i onntially ditributd Th ditribution of th tim that a cutomr nd in th bank Th aociatd tranform. f. 4 4 f th fatr tllr th lowr tllr. 4 4 y y f y y () [ ] 4) (for < dy dy dy y y y y y y y y E cf..
21 Sum of Indndnt Random Variabl Addition of indndnt random variabl corrond to multilication of thir tranform Lt and b indndnt random variabl and lt W. Th tranform aociatd with W i () [ W ] [ ( )] [ ] [ ] [ ] E E E E E () () W Sinc and ar indndnt and and function of and rctivly or gnrally if n i a collction of indndnt random variabl and W L K n ar W () L n Probability-Brlin Chn
22 Illutrativ Eaml (/) Eaml 4.. Th Tranform of th Binomial. Lt K n b indndnt Brnoulli random variabl with a common aramtr. Thn i () ( ) for i Kn If L n can b thought of a a binomial random variabl with aramtr n and and it corronding tranform i givn by () () n n i i Probability-Brlin Chn
23 Illutrativ Eaml (/) Eaml 4.. Th Sum of Indndnt Poion Random Variabl i Poion. Lt and b indndnt Poion random variabl with man and μ rctivly Th tranform of and will b th following rctivly () ( ) μ () ( ) cf..5 If W thn th tranform of th random variabl W i W ( ) μ ( ) ( μ )( ) From th tranform of W w can conclud that W i alo a Poion random variabl with man μ Probability-Brlin Chn
24 Illutrativ Eaml (/) Eaml 4.. Th Sum of Indndnt Normal Random Variabl i Normal. Lt and b indndnt normal random variabl with man μ μ y and varianc σ σ y rctivly Th tranform of and will b th following rctivly σ σ y μ () () μ If W thn th tranform of th random variabl W i W ( σ σ ) y ( μ μ ) From th tranform of W w can conclud that W alo i normal with man μ and varianc σ σ μ y y y cf..8 y Probability-Brlin Chn 4
25 Tabl of Tranform (/) Probability-Brlin Chn 5
26 Tabl of Tranform (/) Probability-Brlin Chn
27 Rcitation SECTION 4. Tranform Problm Probability-Brlin Chn 7
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