Examples for 2.4, 2.5

Transcription

1 STAT 400 Exmles for 2.4, 2. Fll 207 A. Stenov Binomil Distribution:. The number of trils, n, is fixed. 2. Ech tril hs two ossible outcomes: success nd filure. 3. The robbility of success,, is the sme from tril to tril. 4. The trils re indeendent.. = number of "successes" in n indeendent trils. n n n n C ( ), where = 0,,, n. E ( ) = n Vr ( ) = n ( ) SD ( ) = n. Brt Simson tes multile choice exm in his Sttistics 0 clss. The exm hs questions, ech hs ossible nswers, only one of which is correct. Brt did not study for the exm, so he guesses indeendently on every question. Let denote the number of questions tht Brt gets right. ) Is it rorite to use Binomil model for this roblem? Yes. Binomil, n =, = / = b) Wht is the exected number of questions tht Brt would get right? E ( ) = n = 0.20 = 3. c) Wht is the robbility tht Brt nswers exctly 3 questions correctly? 3 ( = 3 ) = = ( = 3 ) = 3 2 = = 0.20.

2 d) Wht is the robbility tht Brt would get t most of the questions right? ( ) = = e) Wht is the robbility tht Brt would get more thn hlf of the questions right (i.e. wht is the robbility tht Brt would get t lest 8 of the questions right)? ( 8 ) = 7 = 0.99 = f) Find the robbility tht Brt nswers between 4 nd (including both 4 nd ) questions correctly? ( 4 ) = 3 = = ( 4 ) = = = Binomil Outcome robbility CDF t n = = ECEL: =BINOMDIST(, n,, 0 ) =BINOMDIST(, n,, )

3 2. An utomobile slesmn thins tht the robbility of ming sle is If he tls to five customers on rticulr dy, wht is the robbility tht he will me exctly 2 sles? (Assume indeendence.) = number of sles. Binomil, n =, = ( = 2 ) = = ( = 2 ) = 2 = = ½. Often On time rcel Service (OOS) delivers cge to the wrong ddress with robbility 0.0 on ny delivery. Suose tht ech delivery is indeendent of ll the others. There were 7 cges delivered on rticulr dy. ) Wht is the robbility tht t lest one of them ws delivered to the wrong ddress? ( ) = 0 = 0.98 = b) Wht is the robbility tht exctly two of them were delivered to the wrong ddress? 7 2 ( = 2 ) = = ( = 2 ) = 2 = = c) Wht is the robbility tht t most two of them were delivered to the wrong ddress? ( 2 ) = 2 = d) Wht is the robbility tht t lest two of them were delivered to the wrong ddress? ( 2 ) = = 0.9 =

4 3. A mjor oil comny hs decided to drill indeendent test wells in the Alsn wilderness. The robbility of ny well roducing oil is Find the robbility tht the fifth well is the first to roduce oil. F F F F S = Geometric Distribution: = the number of indeendent trils until the first success. x, x =, 2, 3,. x E ( ) =. Vr ( ) = A slot mchine t csino rndomly rewrds % of the ttemts. Assume tht ll ttemts re indeendent. ) Wht is the robbility tht your first rewrd occurs on your fourth tril? F F F S Geometric, = 0.. b) Wht is the robbility tht your first rewrd occurs on your seventh tril? F F F F F F S Geometric, = 0..

5 c) Wht is the robbility tht your get three rewrds in ten trils? Binomil, n = 0, = ( = 3 ) = d) Wht is the robbility tht your third rewrd occurs on your tenth tril? [ 9 trils: 2 S s & 7 F s ] S Negtive Binomil, = 3, = 0.. Negtive Binomil Distribution: = the number of indeendent trils until the th success. x x x, x =, +, + 2,. E( ) =. V( ) = 2. ECEL: = NEGBINOMDIST( x,, ) gives ( = x ) e) Wht is the robbility tht your fourth rewrd occurs on your fifteenth tril? [ 4 trils: 3 S s & F s ] S Negtive Binomil, = 4, = 0..

6 f) Wht is the robbility tht your get four rewrds in fifteen trils? Binomil, n =, = 0.. ( = 4 ) = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Let be rndom vrible with Geometric distribution with robbility of success. ( > ) = = = ( ), = 0,, 2, 3,. = number of indeendent ttemts needed to get the first success. ( > ) = ( the first ttemts re filures ) = ( ), = 0,, 2, 3,. Ex. Let denote the number of rolls of fir -seded die needed to observe the first. ( 4 7 ) = ( 4 7 ) = ( > 3 ) ( > 7 ) = For ositive integers nd b, ( > + b > ) = ( > b ) ( memoryless roerty ). ( > + b > ) = b = b = b = ( ) b = ( > b ).