Discrete Random Variables: Expectation, Mean and Variance

Transcription

1 Discrete Rdom Vriles: ecttio Me d Vrice Berli Che Dertmet of Comuter Sciece & Iformtio gieerig Ntiol Tiw Norml Uiversity Referece: - D. P. Bertseks J. N. Tsitsiklis Itroductio to Proility Sectios.3-.4

2 Motivtio (/ A Illustrtive mle: Suose tht you si the wheel k times d tht k i is the umer of times tht the outcome (moey is mi (there re distict outcomes m m K m Wht is the mout of moey tht you eect to get er si? The totl mout received is m k m k L m k The mout received er si is M m k m k k L m k Proility-Berli Che

3 Motivtio (/ If the umer of sis k is very lrge d if we re willig to iterret roilities s reltive freuecies it is resole to ticite tht m i comes u frctio of times tht is roughly eul to i i k k i Therefore the mout received er si c e lso rereseted s M m k m k L m k k m m L m Proility-Berli Che 3

4 ecttio The eected vlue (lso clled the eecttio or the me of rdom vrile with PMF is defied y [ ] C e iterreted s the ceter of grvity of the PMF (Or weighted verge i roortio to roilities of the ossile vlues of The eecttio is well-defied if < Tht is coverges to fiite vlue ( c 0 c Proility-Berli Che 4

5 Momets The -th momet of rdom vrile is the eected vlue of rdom vrile (or the rdom vrile Y Y g [ ] ( The st momet of rdom vrile is just its me (or eecttio is termed s rised to the ower of (or the th ower or the th ower of. Proility-Berli Che 5

6 ecttios for Fuctios of Rdom Vriles ( Let e rdom vrile with PMF d let g e fuctio of. The the eected vlue of the rdom vrile g is give y [ g ( ] g ( To verify the ove rule g g( Let d therefore Y Y ( y [ g ( ] [ Y ] y ( y y Y y ( { g ( y} y { g ( y } y g { g ( y } y y ( (? y y y y 3 4 g ( ( Proility-Berli Che 6

7 Vrice The vrice of rdom vrile is the eected vlue of rdom vrile ( [ ( [ ] ] vr [ ] ( The vrice is lwys oegtive (why? The vrice rovides mesure of disersio of roud its me The stdrd derivtio is other mesure of disersio which is defied s ( sure root of vrice σ vr ( sier to iterret t ecuse it hs the sme uits s Proility-Berli Che 7

8 A mle mle.3: For the rdom vrile with PMF vr / 9 0 if is otherwise 4 iteger [ ] 0 9 ( ( [ ] Or let Z 4 i the [ ] [ ] ( [ ] rge [-4 4] Discrete Uiform Rdom Vrile /9 if z 496 Z ( z / 9 if z 0 0 otherwise 60 vr ( [ Z ] zz ( z z Proility-Berli Che 8

9 Proerties of Me d Vrice (/ Let e rdom vrile d let Y lier fuctio of where d re give sclrs The [ Y ] [ ] vr Y vr( g ( If is lier fuctio of the [ g( ] g( [ ] How to verify it? Proility-Berli Che 9

10 Proerties of Me d Vrice (/ [ Y ] ( ( ( ( [ ] vr ( [ ] Y ( [ ] vr ( [ ] ( vr ( Proility-Berli Che 0

11 Vrice i Terms of Momets ressio We c lso eress vrice of rdom vrile s vr vr ( [ ] [ ] ( ( [ ] ( [ ] ( [ ] [ ] ( [ ] [ ] [ ] [ ] ( [ ] ( [ ] Proility-Berli Che

12 A mle mle.4: Averge Seech Versus Averge Time. If the wether is good (with roility 0.6 Alice wlks the miles to clss t seed of V5 miles er hour d otherwise rides her motorcycle t seech of V30 miles er hour. Wht is the eected time [T] to get to the clss? V T ( v g T ( V ( t 0.6 if v if v 30 [V [ V ] [ T ] V 0.6 if t if t 30 ( V ] g( [ ] However [ T ] [ g V 5 Proility-Berli Che

13 Me d Vrice of the Beroulli mle.5. Cosider the eerimet of tossig ised coi which comes u hed with roility d til with roility d the Beroulli rdom vrile with PMF ( if if 0 [ ] 0 ( [ ] ( 0 ( vr ( [ ] [ ] ( Proility-Berli Che 3

14 Me d Vrice of the Discrete Uiform Cosider discrete uiform rdom vrile with ozero PMF i the rge [ ] if otherwise 0 if K [ ] [ ] 6 6 [ ] [ ] 6 6 vr Proility-Berli Che 4

15 Me d Vrice of the Poisso Cosider Poisso rdom vrile with PMF 0 K e [ ] e e e [ ] 0 0 e e e [ ] 0 e [ ] e e e Proility-Berli Che 5 [ ] [ ] vr

16 Me d Vrice of the Biomil Cosider iomil rdom vrile with PMF K 0 [ ] [ ] 0 [ ] [ ] [ ] o (to e verifiedlter 0 [ ] [ ] 0 Proility-Berli Che 6 [ ] [ ] [ ] [ ] [ ] vr

17 Me d Vrice of the Geometric Cosider geometric rdom vrile with PMF K [ ] 0 (let - < d d d d [ ] [ ] [ ] lter o verified e (to [ ] [ ] 0 (let d - < [ ] [ ] [ ] [ ] [ ] 3 vr - d Proility-Berli Che 7 -

18 A mle mle.3: The Quiz Prolem. Cosider gme where erso is give two uestios d must decide which uestio to swer first Questio will e swered correctly with roility 0.8 d the erso will the receive s rize $00 While uestio will e swered correctly with roility 0.5 d the erso will the receive s rize $00 If the first uestio ttemted is swered icorrectly the uiz termites Which uestio should e swered first to mimize the eected vlue of the totl rize moey received? 0. 0 P [ ] Y P Y ( y 0.5 y 0 [ Y ] y y 300 Proility-Berli Che 8

19 Recittio SCTION.4 ecttio Me Vrice Prolems Proility-Berli Che 9