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1 Probability & Statistics, BITS Pilai K K Birla Goa Campus Dr. Jajati Kshari Sahoo Dpartmt of Mathmatics BITS Pilai, K K Birla Goa Campus

2 Poisso Distributio Poisso Distributio: A radom variabl X is said to hav a Poisso distributio with paramtr, if its dsity fuctio is giv by: f ( ; ) for 0,, 2,...! 0. BITS Pilai, K K Birla Goa Campus

3 BITS Pilai, K K Birla Goa Campus To vrify P(S) = for Poisso distributio formula 0 ) ; ( f Sic th ifiit sris i th prssio o th right is Maclauri s sris for, it follows that. ) ; ( 0 f 0! 0! Poisso Distributio

4 Poisso Distributio E( X ) f ( ) E( X ) Ma ad Variac 0 0 E( X )! ( )! BITS Pilai, K K Birla Goa Campus

5 Poisso Distributio 2 2 ( ) f ( ) E X E( X ) ! ( )! ( ) ( )! BITS Pilai, K K Birla Goa Campus

6 Poisso Distributio ( ) ( )! ( )! E( X ) E( X ) BITS Pilai, K K Birla Goa Campus

7 Poisso Distributio Momt Gratig fuctio tx t M X ( t) E( ) f ( ) 0 t 0! t 0! BITS Pilai, K K Birla Goa Campus

8 Poisso Distributio z whr t z 0! z ( ) M ( t) X t t BITS Pilai, K K Birla Goa Campus

9 Eampl- Th umbr of traffic accidts pr wk i a small city has a Poisso distributio with ma qual to.3. What is th probability of at last two accidts i th t wk? 2//206 BITS Pilai, K K Birla Goa Campus

10 Poisso Procsss Suppos w ar cocrd with discrt vts takig plac ovr cotiuous itrvals (ot i th usual mathmatical ss) of tim, lgth or spac; such as th arrival of tlpho calls at a switchboard umbr of rd blood clls i a drop of blood (hr th cotiuous itrval ivolvd is a drop of blood). umbr accidts i a city pr yar. BITS Pilai, K K Birla Goa Campus

11 Stps for solvig Poisso procss problms Dtrmi th avrag umbr of occurrcs of th vt pr uit (i.. λ). Dtrmi th lgth or siz of th itrval (i.. s). Th radom variabl X, th umbr of occurrcs of th vt i th itrval of siz s follows a Poisso distributio with paramtr k= λ s. i.., s ( s) f ( ) for 0,, 2,...! BITS Pilai, K K Birla Goa Campus

12 Eampl-2 Th umbr of traffic accidts pr wk i a small city has a Poisso distributio with ma qual to 3. What is th probability of at last o accidts i th t 2 wks? As: //206 BITS Pilai, K K Birla Goa Campus

13 Eampl-3 Suppos flaws (cracks, chips, spcks, tc.) occur o th surfac of glass with dsity of 3 pr squar mtr. What is th probability of thr big actly 4 flaws o a sht of glass of ara 0.5 squar mtr? As: 047 2//206 BITS Pilai, K K Birla Goa Campus

14 Eampl-4 Th arrival of trucks at a rcivig dock is a Poisso procss with a ma arrival rat of 2 pr hour. (a) Fid th probability that actly 5 trucks arriv i a two hour priod. (b) Fid th probability that 8 or mor truck arriv i a two hour priod. (c) Fid th probability that actly two trucks arriv i a o hour priod ad actly 3 trucks arriv i th t o hour priod. Solutio: Giv λ = 2 trucks/hr, BITS Pilai, K K Birla Goa Campus

15 (a) s = 2 k = 4 P(X = 5) = 0.56 (b) P(X 8) = - F(7;4) = = 0.05 (c) For first hour priod th probability is f(2;2) = /2! = ad scod hour priod th probability is f(3;2) = /3! = Ths two itrvals do ot ovrlap so th couts ar idpdt, hc rquird probability = f(2;2). f(3;2) = (0.2707)(0.804) = BITS Pilai, K K Birla Goa Campus

16 BITS Pilai, K K Birla Goa Campus Poisso Approimatio to th Biomial Distributio To Show that wh ad p 0, whil p = rmai costat, b(;,p) f(; ). ), ; ( ), ; ( b p b Proof: First w substitut / for p ito th formula for th biomial distributio, w gt )!!(!

17 BITS Pilai, K K Birla Goa Campus p b! ) (... 2) )( ( ), ; (... 2 If, w hav!... 2 Poisso Approimatio to th Biomial Distributio

18 Poisso Approimatio to th Biomial Distributio ad. Hc b( ;, p) f ( ; ) for! 0,,2,... BITS Pilai, K K Birla Goa Campus

19 Poisso Approimatio to th Biomial Distributio Not: A accptabl rul of thumb is to us Poisso approimatio to th biomial distributio if 20 ad p 0.05; if 00, th approimatio is grally cllt so log as p 0. BITS Pilai, K K Birla Goa Campus

20 Poisso Approimatio to th Biomial Distributio Compariso of Poisso ad biomial probabilitis Eampl: It is kow that 5% of th books boud at a crtai bidry hav dfctiv bidig. Fid th probability that 2 of 00 books boud by this bidry will hav dfctiv bidig usig: (a) th formula for th biomial distributio; (b) th Poisso approimatio to th biomial distributio. BITS Pilai, K K Birla Goa Campus

21 Poisso Approimatio to th Biomial Distributio Solutio: Giv = 2, = 00 ad p = 0.05, =.p = (a) b( ;00,0.05) (0.05) (0.95) (b) f (2;5) ! Th diffrc btw th two valus w obtaid is oly Wh w us Tabl 2, f(2; 5) = F(2; 5) - F(; 5) = = BITS Pilai, K K Birla Goa Campus

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