Fo DA

L z/

ProbabilityReview

# 2

Probability Density

Functionsdf )

-

continuous R V X :D → r = IR

f× : IR - IR

Pr CXEA ) = {e. A

fxcw )dwa

eventA er

cumulative density function-

( cdt )

Fact ) = Sw !→fxlw)dw=R HEAT )

c- Co , DAe -

- fast ]fx ( w ) = dT#

Normalkandomvari.at#X-NCu,o)fxlw)--'E*rexpf-YjQ

)exp ( x ) = ex

IIE→

mean

ExpectedVa

MfxR.v. X : A → r - IR

Efe ]

discrete ECX ] = ftp.lw.prcx-wT )& 'm -1continue

= ↳ w .f×Cw)dwStow -1

-

fair 6 - sided die r

-41*3;. . .

%3

RU. X REX -

- i ] = I for i = I . . -6

E =& wi - PRCX = wi ] = 1. I t 2. It . . .

6. IWier

= 2¥ =3 .

5

Linearity of Expectation-

R.V. s X

,Y scalar a

ECXXTY ] = AE t ELY ]-

Z-

Average Height ECHT - ttssm

barefoot herstht fu Hull.to?5o?gm)

Shoe height S--tem/2amµcm/4#b. I 0.1 0.5 a. 3

( oil t Coil )G ) t d.5) 3+4.334

Efx ?-_ECHnioo-SJ-ioo.EE . Dt ECD= 175.5cm + 3cm

= ( 78.5cm

Variance- fixedquantrts

Rv .X

Varcx ] = flex- Effy) )= ECXJ - FEED

'

E -EGAD = E[ x'

-

zx.EC/DtECxJJ--ECx7-zET5ECx3-ECx3--ECxJ

- Efx ]?

-

Rt.

X.

Scalar a war [ xx ]= a' Varcx ]

standarddev.at#Tx=VarCxT

-

Shoes

5--115--2153/5=40.1o .

I ①s5 O - 3

ECS ] =3

Narcs ]= ECG-ECSDJ a i

=⇐&⇒%Cs=i ] - ( i - 35=4.DK - 3)'

+ Called+ Castles ;D't6.3¥'

TX -- ⇐ 0.894 = 0.1 Last o . I to -6.3k

= 0.8

Covariance X,

Y both R.tl.

Cooke . 'D = Eflx - ECXDCY - EGBDq

Joint RV.

-

X, Y

joint pdf fx.yirx.ly → CAND

discrete f*µ(x. g) =Pr(X=x .Y - g)

marginal pdff x

Cx) = ftp.fx.ylxis) - go.dz?rCx-x.Y--D

continuous

fatal = Syeryfx , yG. g) dg

marginal edf Fay ( a g) = Pr [ XIX,

Yes ]

Random X ,Y

independent it Aac , ( x. ytfxk ) .

↳ Cg )

condrtreualdistr-bt.cnfx , ,

Cx ly ) = Pr Cx I ' Ey ]= fx.ylx.SI

fy (g)