Summary Notes

,Chapter 3 :.



eontinious


→ Random Variables .

discrete

→
Probability Distributions .




☐ RV → Probability mais faction .




→
Probability faction
→ ( Rv → Probability density function
→ its
possible values ( ie sample space )
→
Probability of possible values PLAN for all n E 5




Probability function Ipf)
→
pln) 70 for NER

→ É pot ;) =D [sum of all
probabilities is 1)
71=0




ag.IR#ls-umd-nterms-)N--
0
arm




E- "
( sum infinity ]
÷
ar = to




Cumulative Distribution function (step function )



Fln ) = P ( ✗ In ) n ER


for DRV→ For > = E pcni)
ni Es , ni
En


ie .
Sum of probabilities d- the possible values of ✗
,
which are less than
to
or
equal see .

,Expected Value of DRA ( mean )



F- (X ) _-
,€
a ,
ni
plait →
Enpln )
n




of
Expected Values linear Transformation :




E(aX+ b) = a EA) + b

=E(aX + b) (poi ))
71




{ aol.pcnt-Eb.ph ) Ep(a) =/
Ex pcn ) Ese pen)= ECK)
a - + BE
pcn ) .




= a ECX ) -1 b F- (b) = b




Variance and standard Deviation of a DRV .




Var ( x ) - Elt -
Ea ) ⇐ Ge -
F- ( × )Ypcu)


Standard Deviation sdcx)= É

Alternate formula :
F- ( x2) -
(Ea ) ]2
•
Expected value of ✗ 2-
Square d- expected value of ✗ .




Var (9×+5)=-92 Varcx ) Varlb)=o

= El Raab) -
F- ( ax -1bJPY ☒ Variance of a constant is zero .




F- flax its -
-

AECXD #}
F- flax -
AEK )Y ]
F- (a2( ✗ -
ECXDZ )
AZE ( X -
ENT

= a
>
Varcx)

, Moments of a random variable : -




→ the Kth moment about zero is Mk = F- ( ✗
"
)
→ the Kth central moment is Mi = EKX ECXDY -




µ, = M - ECX) and Mz'=Var( X)


Moment generating faction : .




"( c)
'
let §et×pcn)
"
Mx ( t) = E = Mx = F- ( ☒ ) for k -1,2
. . . . . . .




→ convenient tool to derive mean and Variance .




N' (a) = EIX) and
"
M /a) = F- ( x2 )




Vavcx) = E(✗ 2) - (EcxÑ= M (a)
"
-
( MYAT




→ If Milt )= Mylt ) ,
then X and Y have the same distribution .




→ If Y=a ✗ + b


My (E) = ebt Mx /at)

→ If Y= ✗ it Xz -1 . . . . .
+ ✗n


Mutt )=.Mxilt )




Contini cus random variable


infinite number of
→ uncountable values / not necessarily )
→ set of possible values ( sample spae ) . is th . real number R ,
or one or more intervals in R




→
Probability density function (pdf) / probability function of DRA

→ denoted by flu)

→
Pdf are not
probabilities of indivisual values .
flu)¥Pl✗=☒ )

→
P( ✗ = ✗ 7=0 Fx (Probability that ✗ has an exact value is
always zero ]
→
Integrals afpdf give probabilities of intervals Plac ✗ c- b) = [ flat .
da