Probability(2)
YanHuang
Review
• Afiniteprobabilityspaceisdenotedby(", $) where
• " isafiniteset(thesamplespace),and
• $ isafunction" → [0,1] (theprobabilitymeasure)suchthat
+ $(,) = 1
.∈0
Wheneverhearing“probability”,makesurethatyouare
clearwhattheprobabilityspaceis: whatisthesample
spaceandwhatistheprobabilitymeasureonit.
ConditionalProbability
If2 ≠ 4, theprobabilityofevenAconditionedonthefactthatB
happens,$ 6|2 = $(6 ∩ 2)/ $(2).
IndependentEvents
EventsAandBareindependentif
$ 6|2 = $(6).
Anequivalentlydefinitionofindependentevents6 and2:
$ 6 ∩ 2 = $(6) $(2).
• Drawing(withreplacement)twocardsfromastandarddeck.Let
E ={thefirstcardisaKing}
F ={thesecondcardisaKing}
• Drawing(withoutreplacement)twocardsfromastandarddeck.Let
E ={thefirstcardisaKing}
F ={thesecondcardisaKing}
Bayes’s Formula(1)
For any two events 6 and2,
$ 6 = $ 6 2 ∗ $ 2 + $ 6 2 ∗ $(2)
Bayes’s Formula(2)
For any two events 6 and2,
G
$ 6|2 =
G
26
26
∗G(H)
∗G H IG
2 6
∗G(H)
Exercise1
• Twourns:
Urn#1has10goldcoinsand5silvercoins
Urn#2has2goldcoinsand8silvercoins
Firstrandomlypickanurnthenrandomlypickacoinfromtheurn.
Whatistheprobabilityitisagoldcoin?
Exercise2
• Twourns:
Urn#1has10goldcoinsand5silvercoins
Urn#2has2goldcoinsand8silvercoins
Firstrandomlypickanurnthenrandomlypickacoinfromtheurn.It
turnsoutthatthecoinisgolden.Whatistheprobabilitythaturn#1
waspicked?
RandomVariable
ArandomvariableisafunctionJ: " → ℝ (fromthesamplespaceto
thereals)
ExpectationofaRandomVariable
• TheexpectedvalueofarandomvariableJ,denotedbyM[J],is
definedas
M J = + $ N J(N)
O∈0
• Example- Faircointossing:defineX(H)=1,X(T)=0.
Shootingcompetition
10
9
James 45% 30%
Venny 55% 18%
8
27%
17%
7
13%
5%
Whoismorelikelytowininacompetition?
6
1%
3%
5
0
2%
Moreexamples
• Whatistheexpectedoutcomeofrollingadice?
• Rollingafairdice,whatistheexpectationofthesquare ofthe
outcomes?
Moreexamples
• Howaboutrollingadicetwice?
LinearityofExpectation
• ForrandomvariablesJ andP (whichmaybedependent),
Q J + P = Q J + Q[P]
• Moregenerally,forrandomvariablesJR , JS , … , JU andconstants
VR , VS , … , VU ,
Q VR JR + ⋯ + VU JU = VR Q JR + ⋯ + VU Q[JU ]
Betterway
• Expectedoutcomeofrollingadicetwice?
ExchangingGifts
• AtaChristmasparty,X friendseachboughtagiftboxandmixedthem
together.Later,eachpersonrandomlydrawagiftboxfromthepile.
Onaverage,howmanypeoplewillgetbacktheirowngift?
ExchangingGifts
• AtaChristmasparty,X friendseachboughtagiftboxandmixedthem
together.Later,eachpersonrandomlydrawagiftboxfromthepile.
Onaverage,howmanypeoplewillgetbacktheirowngift?
ExchangingGifts
• AtaChristmasparty,X friendseachboughtagiftboxandmixedthem
together.Later,eachpersonrandomlydrawagiftboxfromthepile.
Onaverage,howmanypeoplewillgetbacktheirowngift?
0,
JY = Z
1,
the \ ]^ person gets back his/her own gift
otherwise
BinaryrandomvariablelikeJY iscalledanindicatorrandomvariable.