Math-> Probability

Recall

Det for a discrete r.v I the p.rs iitymessfen.Cp.zE px

Ca

px Ca EetpCE a t only defined for a in Range

Def The cumulativedistributiont cc.IE for a r v E F It is defined as

felt Meet It can be any real

Et Flip a coin 3 times Let Hs flipped

Range 0,42 3

d find the full pint and plot

PECO L p l 3

8 8

PE D p 3 I s 8

pEia

y

318 as g p a

48 Yg

It Is a

e F.n dthd.tn and plot F Hf Meet

Start by finding adf.at Range

FILO P EEO _PCE g Yg

fell PCEED PCE o PCE D Kg t 31g 4 8

FE 2 P CEE 2 PCE D t PCE D tPCE 2 Yg t 31g t 3 8 7 8

FEC3 P CEE 3 I

FE 1.34 p CEE l 34 PC EE D 418

FE12 7 p CEE 2 7 p CEE 2 7 8

E It PCIe highest in Range

FEC too P EE 100 O

FE too P CE E 109 p CEE 3 1

q

i

y y Felt

I

48

J PCE D III i I

b L I t

graphof the

FAI The c df for a discrete au will look like a staircase A step at E K means K is in the Range of E The step size at K is pCE k

EI Roll two 4sided dice Let I minimum of the two

s rolled

e.g 3 d E 2 4 4 I 4

Find RangeI p.m.f c d f of E Range I 2,3 43

p.m.f.cl I 2 3 4

PE a 7 16 546 6 46

T PCE D PThe min rolled is 1

lb 1 142 12,1 l 3 13,1 H 14,1

PCE _2 PThe men rolled is 2

is 94,7

felt 41 I Etc 2

u

yet

FC2 IS P CEE d 1 P CEE 2

P D t PCE D 74ft 546 1416

y Y y f lt

g PEH7116 I 416 a 8 8 16 o

44

I l l l h P I l l t l L 3 4 I 2 7 4

propertiesofc.d.fi p.n.f l c df is a nondecreasing fan

If a Eb Fca c Fcb

2 OE Flt E 1 for all t OE p a E 1 for all a in RangeE

3 1 F P CE t

F I PCEet PCCE.tt P I t

4 If a b then Fib Ffa Plas b

i r FCbt P b

a b fCa PCEa

5 lineEfH 0 liar 1

6 Let flkt l.ie Flt C k

FCK't _linflt t 5kt

Then Flk'T FCK fn.ifneisght.ijts.is

f Kt FCK P E k

In above Ex F 3 Ha FC3t 15ha FCD

f 37 f 5 1546 1246 3

4 P 3 I

A Spla inRange

L

g

8 Felt Sp K Ket in Range I

e g If Range 7 31,23,43

F 2 4 p 1 pls

EI Given the graph of the cdf of below

y y Felt

l

ii i iii i

s l L b f f f t

Find Range ftp.m.f of E

Range E El 2,4 5,63

nattiest

ss2.75Expectationtker.ua

MotivatingEx's Flip a coin If we flip It win 10 If we flip T vis 0

Scenario If the coin is fair we expect to win on average

Scenario If the coin is biased w PCH ko PCT

we expect to win on average 1

Det If E is a discrete au w p.m.f.pe Ca then the expectation or expected of E is F E ape.ca Exp x

x ina in Range Range

Exe Flip a coin Win 10 for H

0 for T Let E winnings in

Find ECE if a coin is fair b coin is biased P Yo

a Range 50,103

E.at EECI3lo pl0 tllDpCl0

o k t lo Ya I

b Range E 0,103

ii F I o p

O t Ilo p lo

o t lo ko