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} . Discrete Random^ Variables^ ( 0kt^ ZTE^ HIF)
* Sx :^ the^
range
of X
the set (^) of all^ possible values^ of X
* discrete RV .^ :^ Sx is a countable set .
god
't
- (^) Probability Mass^ function^ (^ PMF^ )^ :^ Px^ I^ x^ )^ = PfXm=x] HEIKE Ex (^).^ 3.^ 5) Two free shots (^) , X :^ the #^ of (^) points Sx =^40 , I (^) , 2 } (^) ② X 0 I 2 g ① (^) pun -^ -^ I^ PNM I I & a (^) samplespat's "
orange ,^
4¥! '÷÷
, elw^ I
-. (^) - - -. - (^)..
O I 2 K
( p )
Bernoulli R (^).^ V^.^ :^114 Hbk^ 'VE (^) an WEE^ ' EE
Sx -^ - to (^) , It 17×64=1 tpp^ , K ' of. sub
experiments
with two^ possible outcomes
,^ at^
: Bernoulli trials
O (^) , elw I (^) p ) It Geometric R^. V^.^ :^ 21¥^ event^ HINT^ Mnt^ H^ Hbk^ ELF^ ,^ Sx^
P
, I^ x^
, = f P ( I^ - P ) " "
a- I (^) , 2,.^.^. (^0) , elw ( n^ , p ) Binomial RV^.^ :^ k^ successes^ in n^ trials
Px ( k) =
f ( (^) I ) Pk^ Ctp^ )^ " (^) "
k=o (^) , 1,2 (^) ,^.^.^. ,^ n O (^) , elw
I n^ , K^ )
- (^) Pascal R. V^.^ :^ Htt^ event^ kik^ KHANH^ Alby^ Zdf
x
Px I a^ ) =
f I KI^ ) (^) pkltpl
, x=k (^) , KH (^) ,^.^.^. O (^) , elw
( k^ ,^ l^ )
- discrete^ uniform RV^.^ :^ equiprobable outcomes
PNC n^ ) =
{ HCl -^ KH^ )^ , n=k (^) ,^ KH^ ,^.^.^. ,^ l
O
, elw
- PoissonR. V^.^ :^ Natatorial^ event^ KITE^ the p, = ya " e
- Ha!^ i^ "^ ⇒^ ' '^ '^2 '^ '^ "
d =
II time interval
O
, elw
average rate
arrivalshed μ 3.4 (^).^ Cumulative^ Distribution^ Function^ (^ CDF^ ) Film =^ PCXE^ x^ )
Ex 3.^ lb ) CDF of
geometric (1/4) R.V.^?
Py Ly ,.
. { (1/4) (3/4) 't '
, y
- (^) 2,3 , - (^) - - O (^) ,^ elw Film
II Pay ,^ = FI ,
GET^
= if =L - ¥ " integer : (^). Fycnyy = f
0 , y s^ I
real (^) I - ¥ , y z , floor function
Averages and^ Expected^ Value
* EH
)=μ×=¥s×x
ex ) Bernoulli^ :^ ELM^
p =p
ex (^) ) geometric : ECXI^
Pll - pl
" (^) = Yp ←
power
series
ex ) Poisson^ :^ EIA^
= II. xd^
e
%!^ =^ a^ ←^ total^ sum^ of^ PMF
ex ) binomial^ :^ EH^ )^
- Eon P' ' Aptn - " = rip I
ex ) Pascal : ECM =
Eif pka^
kyp
- MSE (^) ( Mean (^) Square Error^ ) → (^) e = (^) ELK E)- 2) ① the^ minimum MSE^ estimate^ = (^) I = (^) ELX ) ② the^ minimum MSE^ =^ EKX-^ EA)^ )^ ' ] =^ Var^ CX)
- (^) Varix ] = ELM
- (^) nth moment =^ EH^ " ] →^ 1st^ moment^ =^ THE nth central^ moment^ =^ El^ # Y^ ]^ →^ 2nd^ central^
moment =^ Hit
R (^) R R (^) R R (^) R O (^) O (^) O O O (^) o X (^) x x x (^) ×
IB
Cbl (^) Px IN -^ -
g
Pll
pl
, a- I (^) ,^ -^ - ; 5 ( I - p ) 5 I
o , ecw
cc)^ (^ I^ - pH N (d) ( I^ -^ 0.9^ )^ L^ o^.^02 10 "
i (^). n Z 2
Ca)^ PL, (d) =^ (F) (^) ,^ f-^ 91.^. ' (^) " { (^) o ,^ elw
(b) PCW -10 ) =
(f)
to
Pascal (^) R (^). V (^). ( kilo^ )
i. pwcws = (
wt )
fEY
"
w.io , " ,^.^.^.
10 - I l { o , eh SIC (a) Puny
k= 3. 4.
{ o (^) , elw gtw ) Kd^ Rtm IHH 't (b) Pwlwi
-11791- 't'T^ , a- (^) on .z
w =3 (^) ( EY (^) ) O (^) , elw ' " kai-yletek.gs#e.ii:i!au 0.5 (^) ,^ l^ =3^ ( EU ) O (^) ,^ elw
Pxnlkl =/na^ ) p 'll^
- pH? x -^ - on (^) , (^2) ,^.^ - in { o (^) , elw Eun ]
II.
x P'
a - pin
=
saints x. . P
a -
p "^
n (^) n. ( n - I (^) )!
= I -
⇐ I @YHkn-p.ca^
P
' P'
Ctp) " - Htt = npE.it:^ P' " HPT
, a np.E.it:7/p4i-psn- " " = np
. II. (^) Pain = np ELM = IE KPCA-kf-PCADtz.pl/=Dtz.PCx=3It--=pCX=D = PIX >^ oft PIX >^ Dt PIX^ >^21 t^
- " t (^) plant PH 's^ ) (^) = II.oplxsk) tplx-3HPH-htpcx.SI
(a) ¥= Emo (^) : (^).
p
- Tomo PR (^) ( r^ ) = g Tomo (^) ( I^ - Tomo) " (^) "
,^ 2=1^ ,^ 2,^
- -^. O (^) , elw (b) (^) EIR ]= I ome Efw ]^
= Ef5mRT= 5 MEIR) = 500
A- I^2 3 (a) (^) ° × Ox^ Ox^ Ox^
. (^) ' Px (^) I x (^) ) = f ( (^) tf ) (^) of " ,^ A-^ I.^ 2.^.
- -. O (^) , elw ( (^) b ) (^) X -^ - I →^ 7=1 m
X=2 →^ F- 3m
} T = 3¥ 11=3 →^ T =^ 5ns
1000T - I
÷
Pt =
( I - G)
q= , t =^ Im^ ,^ 3m^ , 5M (^) ,^.^.^. O (^) , elw ord : Pr ( H=^ 0. "
r -^ - k - to
{ I^ -^ 0.^ "
r= -^ to
° (^) , elw ultra :^ PRIM^ = g
,^ r=^ K^
0 , elw
ELR)^ ord - ECR)^ ultra = (^) ( k-1070.
t (-1014-0.910) - f Ck^ -3070.95 't^ (-3074-0.950)}
Tff
mm run^ mm 0.65 0.60^ 0.
0.25k^ I^ - t
. :^ KC^80 ⇒^ ordinary , k^ >^
80 ⇒ ultra -^ reliable
k
Pk (^ k^ )^ =^ X^ e-^ 4k^!^ ,^ k=0^ , 1,2^ ,
{ o
, elw Elk (^) ] = EEK ' -4k^ ake ! = DIE
the -4*1 (^) ,! =^ x
K -1=
Varlk) =^ Elle^ )^ -^ (ECK))
! Elk C Kt ) ] TECK ) - LECK] )
' = (^) data - as =^ a
ELKCK - H) = II Hk^ -^ Dake^ -4k^!^ =L^ the^ -4*-2 , I^ =D^
K -2=
