,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
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
