ページ1:

設汽車在才一道門.
Car
p(s) = p (D₁) p($10₁) + p (D₂) pl$1D2)+p(D3)p($103)
= 0 + = 1 +
1 = 3/1/2
> car
1+1=
3 car
|| (7) = .
(不連續) (布)
Ch 2. Discrete distributions (2600)
(r.v.)
2-1: random variables of discrete type.
XIR (a real-valued function)
2-2: Mathematical expectation $1.
2-3: Special mathematical expectations.
eta
variance & Var(x) = E [(x-(x))"] = 0x 5x = √5=
moment-generating function (4) 32 ± 2 (m.g.f on M.G.F)
m(t) = Ele
characteristic function 11/12 (ch. f.)
$ (+) = E (etx) i = FI
|| 8(+)| = | Electx | | ≤ El ex | = |
probability generation function. In 145x32 (p.g.f)
4(+)/p(t) = E(+")
2-4: The binomial dist. = 1&13+)650
二
2-5: The negative binomial dist. = 11500
2-6: Poisson dist. tasu.
Have a nicetime

ページ2:

No
-1D3)
Have a nice time
Def: A random variable (r.v.) X
X: SIR 2: outcome space.
X (s) = x, SE, XEIR.
(Support)
m
PX = x
Date
Space of X = {x:X (A) = x, ^=SL, XEIR} = S
eg: Toss a die
52 = { 0, 0, 0,
X = the number of dots.
X(四)=1
X = 1
X(四)=2
X = 2
間
space of x = {1, 2, 3, ..., 6}
x() = 6
記x=6
Y= the
Square
root of the number of dots.
Y(0)=5T
Y(0)=52
space of Y = {51, 52, ..., 56}.
Y()=56
Def: probability of mass function of X (p.m.f) In 14 F 2 J, Z
X: discrete type of rv.
f(x) = p.m.f of x, satisfies (a) f(x)>0, XεS
(b)姦f(x)=1.全机率
y os finst
(c) P(A) = P(XEA) = x Af(x) if A≤'s
As X is discrete, pmf=pf RP f(x) = P(x=x). pf: prob. function..
Def: (cummulative) distribution function of X ((c)d.f of x)
F(x) = P(X ≤x) = x(t)
GEE-JUMP

ページ3:

eg.
Toss a fair die
Date
X= # of dots.
space of x = {1,2,.,6}
Find the pmf of X and cdf of X.
sol; f(x) = 1 = x=1.2..., b. pmf.
= 1 +
o, elsewhere.
F(x) = P(X ≤x)
-|
P(X ≤ -1) = 0
(x~ discrete uniform)
0
No
Have
nice time
f
pmf
Step function.
9
1
1.5
F(x) = 10
亡
[x]
6
X<1
[x] ≤ x < [x]+|
1, X36
123456
1: X is discrete.
cdf of X is step function..
123456
cdf of X is right-continuous
0≤ F(x) ≤
eg. Roll a 4-sided fair die twice.
X = the maximum of two outcomes.
Outcome space √2 = {(d₁,da): d₁= 1.2,3,4, d² = 1,2,3,4}
Space of X = {1,2,3,47
Have a nice time

ページ4:

Have a nice time
f(1) = p (x = 1) = p({ (11) }) = 16
3
f(2) = p (x=2) = p({(1, 2), (2,1),(2,2)}) = 1/16
I
Date
No
2x4
2 3 4
3.
f(x) = {1, x = 1,2,3,4
52
0
elsewhere.
₤13) = &
+(4)=76
Ex: cdf of x.
16 16 16
Ni
2 classes, I & II.
IL
N₂
• Ten objects (n≤min (N₁, N₂} )
X = # of objects in class I among ʼn selected objects.
space of x = {0, 1, 3, n}.
2,...,
f(0) =
(N)(NZ)
h
(N₁+ N₂)
f(1) =
(M)(二)
(NTN)
n
pmf off: f(x) =
Ni+N₂
X = 0, 1, 2,..., n
X ~ hypergeometric (N₁, N₂ ; n)
check: fix) = 1
号(x)(3) (NM+2)
n=0
Ni+Nz
h
+
(M+N₂)
0)
eg: 100 fuses / 20 defective (7 Bos)
so good.
5 fuses are selected at random.
X = # of defective items in 5 selected fuses
Find the pmf of X and its space
Space
of X = {0,1,2,3,4,57, X-hyper (70, 80; 5)
fon = (*)(8)
(189)
X=0,1,2,3,4,5
GEE-JUMP

ページ5:

Date
eg:
fix) is a pmf of X.
(1) f(x)=(x, x = 1, 2, ..., n.
f(x)=1 c₁(x) = 1 C = {n+1} #.
n(n+1)
(>) f(x) = (x², x = 1, 2, ..., n.
C = n(n+l)(n+1) #.
Ex. (3) f(x) = cx³, X = 1, 2,..., n.
C =
d
(4) f(x) = x, x = 1, 2, …---
= d = X
d. (= divergent.
d
(5) f(x) = = X = 1,2,--
d
b
d
=
TU
(6) f(x) = (x+1)(x+2), x=2,3,4...
= d = 3
d [ 21/2/2 (x+1)(x+2)] = [
Have a nice time
No.
Have a
2-2

ページ6:

No.
Have a nice time
無看又实驗結果
2-2 (Mathematical) expections.
Def: X is a discrete- type v.r. with pmf f(x).
E[u(x)] = u(x). f(x), if Elu(x)/coo.
E[u(x)]: expected value of u(x)
(mathematical) expection of u(x).
(arithmetic) mean of u(x).
eg: Toss a balanced die.
X= # of dots.
Date
E(x) @ E (x²³) E(e*) © E(lux) © E(10x+3)
-Sol; f(x) = { {, x = 1, 2, ..., 6.
o elsewhere.
_ @ E ( x ) = 1 · — — + 2 · — — + ··· +6.1 = 21 = 1
+.....
© E(X) = = = (1 + 2 + - +6³) = 21/
E()=(e+e+-+e)
2
@E (lu x) = = ( lul+...+lub) = ln 120.
E (10x+3) = 121 (10x+3). f(x)
=100巷1(x.f(x)+3f(x)
= 10.E(x) +3.
Thm: ( Properties of expectations).
(a) E (c) = C, C is a constant..
(b) E[cu(x)) = C.Elu(x)]
(c) E [C₁u, (x) + C ₂ U₂ (x)] = C₁E (U, (x)) + (₂E (U₂ (x)
(d) If uxo. then Elu(x)] >0.
(e) If u(x) > V(x), then E (u(x)) > E (v(x))
(c)
U(x)-V(x) > E (u(x)-V (x)) = E (u(x)) -E (v(x)) >0.
GEE-JUMP

ページ7:

No.
Have an
E(X)=5xfx)
17 E(x) = x
eg. f(x) = x, x = 1.2.3.4. is the pmf of x
E(x), E(x²), E (x(5-x)]
4
.
Date
E(x) = $ x ⋅ fix ) = 1 · 1 6 + > · 7/16 + 3 + 1/16 + 4 + 1/16 = 3.
E(x) = x* f (x) = (1+2²+3+4)=10.
Ex-x)]=5E(x)-E(x) = 5.
eg. Assume E[(x-b)] exists, b is not a function of X.
Find the value of b sit E [(x-6)²] is minimized...
E[x-b]=E[x2bx+b²]
= E(x²³) - 2b E ( x ) + b² (b)
h'(b) = 2b-2E(x) = 0 + b = E(X)
1(b)=2>0
-[[(X-E(X))"]
As b = E(x), E((x-6)²] has a minimum
(n-x
✓ eg. x ~ hyper (N, N₂ ; h), Find E(X) (f(x) = x=0,1,--, n)
E(x) = x.
()()
(M+M)
x=1 (N,+N₂+N+N
(x)=(x)
N
y=x-1 = n
N+N
n-1. h
(N + N = -1)
n-1
Have a nicetime
= n.
N
NIN₂ #

ページ8:

No.
Have a nice time
Date
X = # of Bernoulli trials needed to obtain the 1st
Ę
P(success for each trial) === p
success.
Find pmf of x
@ E(X) = +
space of x = {1, 2, .... }
• f (x) = (1-p)" " p, x = 1.....
2,
10 elsewhere.
✓
E(x) = x (1-pp
= ·
1
A = 1 + 2 (1 - p) + 3 ( 1- p)² + .....
-(1-p)A = (1-p) +2 (1-p)+
PA = 1 + (1 - p) + (1 - p)²+
=
.....
1-11-1) = + = A = 1/2
X~ Geo (P).
X has the geometric dist. with p(success) = p.
eg. X ~ Geo (to) = E(X) = = = 10 #
eg. Flip a fair die. Find the mean number of flips.
to obtain 6 different numbers.
\ + ½ + ½ + ½ + ½ + ½ = 14.7 #
GEE-JUMP

ページ9:

Date
No.
2-3 Special mathematical expections.
Def: Variance of X.
(expected value of x)
Mx-Elx) mean of X
0x = Var (x) = [(x-μx]] = x + (x-μx) - fox if exists.
St. dev.
Def: standard deviation of x.
註:
x = Var(x)
: X30 = E(X) > 0, Var (X)>0, 570
2
Theorem: Var(x) = E(X) -μ²
pf: LHS = E [(x-μM)²]
= E [X² = μx + μµ³]
Theorem: abEIR
= E (x²³) -> ME(X) + μM²
2
= E(X) - μ² = " RHS.
Var (ax+b) = a var(x)
pf: LHS = E { [ax+b= (aμ+b)]"}
=E[a²(x-μ³]
=
>
• a² E [(x-μ) ³] = a var(x) = RHS.
: Var(b)=0, bEIR.
_ @ E(b) = b
E(ax+b) = a E(x)+b.
Have a nice time

ページ10:

10.
ue of x)
Have a nice time
eg:
: Var (x) = 8, E(X) = 2.
Date
No.
Find El-2x+5) Var(-2x+5) E(X) @ E(5) © Var (5)
DEL-2x+5)=-2.2+5=1
© Var (-2x+5) = (-2)²- 8 = 32.
© E ( x ) = Var(x) + μ² = 8 +4=1>
(5)=5
Var (5)=0.
Def: the rth moment of (the distribution of) X about the origin.
E(x) = sx f(x), if exists. r=112.....
XES
St
13: E(X) = 1st moment of X.
E(X) = 2nd moment of x
E(X) = 3rd moment of x
階乘!
Def: the rth factorial moment of the dist) of X.
r項
I
E [ x (x-1) (x-2)-(X++1)] = xsx(x-1xx-2)... (x++1)· f(x)
XES
13): E (x) = 1st factorial moment of x
nd
E[x(x-1)] = 2" factorial moment of X.
(NONE)
=
Var (x) = E(X)-μ
=E[x(x-1)] +E (x) -μ².
= N₁ (N₁-1) = (x-2/ \n-x)
eg: X ~ hyper (N₁, N₂ ; h)
DE (X) = ".
N₁
Ni+N
✓ Var(x) =
N(N-1)/N1-2)
E[X(X-1)] =
XTX).
(xx\n-x)
y=X-2
Var (x) = n
N₁+Nz
N₁+Nz
:) (
1
N₂
(M-2) (N2)
= (MAND)(NANGI) (NIANZZ)
1-2 (1-2) (1-2-4
N(N-1)(-1)
(NN) (N,+N₂-1) Y=0
N
NI+N2-Y
1-2
h(n-1)
GEE-JUMP

ページ11:

No.
Def: the moment-generating function of X.
ho, llch Elet*) exists.
+
Date
Then the function of ± defined by M(+) = E(**)
M(t) = E(etx) = x+s ex. f(x) is called the m.g.f of x.
2x-1
eg: f(x) = 1 x 6", x = 1, 2, 3, 4
0 elsewhere.
M(+) = E(etx)
= 1·16 +
at 3
Theorem: (properties of mgf)
Find the m.g. f of x.
4t7
16
+
e.
16
(a) X and Y have the same distributions.
<=> Mx(t) = MY(+) (Uniqueness of m.g.f)
(b) M(0)= |
(k)
(c) M (0) = E(X) k = 1, 2,..., if E(X) exists.
M (+) = E ( * ) = E [ = x + 1] =
M(t)=E(克)=E[嚚均
E(X)
k
x +
(31): M10) = E(X) M'(0) = E(X) (M" = E(X³)
eg: Mx(t) = eft Find the pmf of x.
fix=f
x=8
10, elsewhere.
t
eg: Mr(t)= +0.
fr (y) = 1 ½, y = 0
13, y = lu 7
・ 0, elsewhere.....
Have a nice time
-
1 at 15t
eg. Mε(t) = je + Je
Since Mz ( 0 ) = 3 + 1/3 + 1
M₂(+) is not a mgf.
Have a

ページ12:

Have a nice time
-t-5t
eg: Mu(t) = + e + + ½ + ½ *
space of U=f-1-5, lu3}
X ~ Geo (p). f(x)=(l-p). P, X = 1. ²......
E(x) = =
+
\ @ Var (x) = E (x² - μ² = E [x(x-1)] + E(x) — μM².
E(x) = 1 / 1 x (1-p) p = 1/2 · p = +/
X-
E [x(x-1)] = =₁ x(x-1) (1-p) p
X-2
= (x =₁ x(x-1) (1-p) ^ Jll-p).p
=
· 13. (1-p) · p = (1-p)
√ ⒸM(+) = E(etx)
=
tx
+ {
(1-1) P
[ep]}
et(1-p)
+
I-P (= e^\l-p), e (1 - p) < |
Pet
Date
= 1-(1-plet, t< -In (1-p)
No.
無記憶
Memoryless property - p(x> a+b/x>a) = p(x>b), abEIR
若x~Geo(P)則入只有無記憶性質,
a
p(x > a) = 1 - p(x≤a) = 1 - 1 1 (1-1)* p = 1-1-(+) - R = (1-1)^
LHS=
P(x>a,x>a+b) P(x>a+b) 11-119+b
P(xx) = (x32) = (1-3)^ = (1-p)² = p(x>b) = RHS.
GEE-JUMP

ページ13:

Date
No.
CCCC
Have a nice
eg:X:
1: X= # of people
that
you
have to ask to find
X-Geo
as
you
are
Someone who was born in the same month
(Assume that each month is equally likely).
Find DE(x) = = = =12
@Var(x)=
p(x >15) = (1-3) 15
(x75\x10) = (1-5)'s
p(x340|x>10) = (1-1/1) "0
eg: Mx(t) = = =² + < luz, Find the dist. of x.
By the uniqueness of m.g. f x~ Geo (s)
2-4: The binomial dist.
success 6x2
Bernoulli experiment - 2 outcomes (failure * BX
✗=
P(x=1)=P
D. KAY, p (x=0) = 1-P.
X~ Ber (P) E(X) = P_1.p=0· (1-p) = !
b'lip)
Var(x) = p(1-p) F(x) = [[(^)] = ( ~ p+oll-p) = p² = p-p²
M(+) = pe² + (1-p) Ele") = e *pte²°(l-p)
+= Y = # of successes in n indep. Bernoulli trials,
each of which has the same prob. of success, say f.
Y~b (rp)
n-y
(p.m.f). frly) = Cy p" (1-1), y = 0,1,..., n.
n-y
check: 1 (y) p³ll-p)^^ = [p+ (1-p1]" = 1
By = z
Have a nice time
| p(x=8)=

ページ14:

d.
= you are
Have a nice time
E(Y) = np....
Var (Y) = hp (1-p)
Date
No.
Mit) = Ele² = (
y=0
n-y
n-y
= 1/170 (1") (pet) (1-p) ^ " = (pe² + (1-p))"
Mit) = h [pe² + (1-p)]" pet ⇒ M'(0) = hp = E(X)
M'(t) = n (-1) [pe² + (1-p)]" (pe^) + n [pet (ip)] - pet
M"(0) = n(n-1) p² + np = E(X)
th
Var (Y) = E (x³- [E(x)) == \(^_^)p²+np-hp² = -hp+np=np (1-p)
eg: Mx(t) = (0,15 e²+ +0,25)
eg:
p(x=8)= p(10-x=>)
= p(10-x≤2)
Find the dist. of X: X~b (12,0.75)
E(X) = 12x0,75=9
③Var(x) = 9x 0,25 = 255
X = # of seeds that germinate in 10 indep, trials.
p(germinate) = 0.8.
11)p(x≤8)=
( p (x = 8) =
“不会發芽.
X~b (10,0,8) 10-X ~b (10,0,2)
10-X
0.87012
0 (x²) 08 % 1 * p (x≤8) = p(10-x>>)
(2) P (X = 8) = ( 1 ) 0.8%
-p(10-x≤1) (3) E(X) = 10.08=8
0,2
(4) Ox = √8.0₁ = √1.6 #
註:(1)Y~b(nmp),Y=成功的次數.
"
= 1-p(10-x≤1)
=1-0.3758
h-Y= 4 x R x = n-Y ~b (n, 1-p)
relatively small.
b (n, NA+N=)
N
· bln
N₁+N₂
(2) N₁, N₂ are large
is
hyper (N₁, N₂; n)
N₁
M = N₁₂
Var(x)= N
N₁
Var(x) = h =
NITN₂
GEE-J
Ni+Nz
NI+N₂ NITNEY

ページ15:

D
Have a ni
CCCCC
A exercise
mmm+1
(3) X1, X2,
+3
No.
identically independent.
b(1,p)
· Xn iid. Beri (p)
Y = X + X2 +
......+Xn(Y=n次实驗几次成功)
To
→Y ~b (n,p)
眾數
(4) The mode of the dist. of Y, Y ~ bcn,p)
高斯.
fim
31
符号.
D
femi)
₤(m) 711
f(x)
n-y
fr (y) = (^) P" (1-p), y = 0, 1, 2, .-, n
[[(n+1)p], (n+1)p & Z
(n+1)p-1 x (n+1)p, (n+1)PE Z
2-5: negative binomial dist. (Pascal dist.)
Y = # of Bernoulli trials needed to obtain r successes
P(success) = p.
-, (y-), y
Y~NBLY P
fr (y) = (1-1) p² (1-p) · P
4+
K-1次成功
y-r次失敗
4th XID
y-rrr-1
y-r
(-) (P)-P.P
y-r.
= (+-) P (1-p), y=r, r+l, rt2....
Recall: Maclaurin's series expansion.
h(W) = & wiseries expansion
== "ki Wk | w/cl
Eh (w) = (1-1) = 0 (k+(-1) wk - negative series
(= 30
check: (-1) P² (1-1)* = |
k+r-11
Ek=y-r [ ("K") (L-p)")] p² = [1-11-p)] = p² =
1
Have a nice time

ページ16:

b)
Have a nice time
E(Y) = (1-1) P² (1-1)
Date
y-r
I
J. ("-") P' (1-p)
y-r
y-T
= r · √ √ =³r (~1) P² (1 - p)² +
Fik = y-r = p² = (+ + (x+1)-1 ) (1-p)"]
k
= r.pt [1-(1-p)] (r+1) = ✗
ㄓㄓ
r(r+i)
1-6(6) 6
r(r+1) (r-1,
E[Y (Y+1)] = √ (y+) (+) P (1-p)
y-r
= (x+1) | | = (1+1) (1-p)
K+(r+2)-11
k = y-r = v(x+1) | | (+(-1) (1-p)^\]
k
(+1)
= r (r+1)P" [1-(1-p)] =
Var (Y) = E[Y(+1)] - E(T)-[E(Y)]"
r(l-p)
=
et(s) tr
e
ty
·1(171)
y-r
My(t) = E (et) = ₁et (1) P(-)
=
y=Y
tr
et pr. f (-1) ["\\-p)]
y-r
k=y-r = (pe²). (kk) [ \\-]+] <
K=0
+
= P(+)" [1-(1-p)²+] (\-p)e* </
Y~ NB (r,p). = E(Y) = ₤
Var (Y) = HH-P)
Pet
No.
X~ Geo (p) = NB (IP).
M(t) = [1-11-ple³], [1-ple < | (+<=lm(1-p)
Ì: (1) X₁, X2, Xr Iid Geo (p)
令Y= Xi + X2+...+ Xr (r次成功实驗次)
→Y ~ NB (rp)
E(N) = E (X₁ + X 2 + ··· + Xx ) = E ( X ) + E (X 2) + + E ( X ) = =
GEE JUMP

ページ17:

Date
No
eg: Mylt) = [100 et ]. Find
1-0.30
the dist. of Y: Y~ NB (5,0.7)
③E(Y)=
5
5.0.3
Var(Y) = 0.09%
罰球
eg: X = minimum number of free throws to make a
total of lo shots. p (shot) =0.8.
DX~NB (10,0.8)
@p(x = (2) = ("a") (0.8) (0.2)
© E ( x ) = = 10
x=10(1-0,8)
10.8) 2
eg: Given E(X) = 0.8, r = 1,2,3..-
Find the dist. of X.
E(x)
k
Mx(t) = = k!
0.8
10.2, x=0
=
f(x)=
0.8,x=1
elseshere
=
= 1 +0.8 (et -1)
ot
=0.2e+0.
eg: F(x)=>", Y= 1.2. Find the dist. of X...
Elx k
M(+) = k=0 ki t
=
=
0
3k k
= kit
3t
est
(at)k
f(x)=11x=3
To elsewhere
fw=L
Have a nice time
CCCCC
e
↓

ページ18:

Have a nice time
普松
2-6: Poisson dist,
Date
No
X =
In
an approximate Poisson
process,
the number of changes
that occur in a given interval.
Approximate Poissom process satisfies:
=
(i) # of changes occuring in honoverlapping intervals are independent.
(>= X 11 X 2.
(ii) P(one change in (w, w+h)) = λh, 7>0.
(iii) P (at least 2 changes in (w, wth)) = 0.
已知單位時間,發生入次
n等分,使得每一等分中,最多
發生一次,每等分長度方
P (one change in (w, with)) = A
x ~ b ( n )
lin
h-200
(2)(六)(1一六)
M-X
lin (n-1)... (n-x+) **
4900 nx
X!
°
**
(x)=
h(n-1)(n-2)(-x+1)(n-x)!
x! (n\x)!
(1-A)" (1-4)
=
✗!
X=0,1,...
X~PIA)
ex xx
f(x) = x^x = 0, 1, 2,
00
E(x)=x
=λ
(check:
0
you 4-6)
X=1 (X-1)! = ^ y = 0 4 1 = 1.
Var (x) = E (x(x-1)] + E (x) = [E(X)]² = \\_
E[x(x-1)] = x(x-1) xx
=D
y=x-2 = λ² [ 1-2 (X-2)!
-
(X-2)
X!
=
I tx e\"
一入 de
✗!
= e
00
I
Thesx
X = X !
tiến
M(t) =
= e.e
= x²
GEE-JUMP

ページ19:

A
DVD
Warra
!
: (1) As A >10, Poisson
n-300, p«<
Normal dist.
(2) X ~ bln.p) ^ hp = 1 > P(A)
lin (1) P (1-P)
n-0
Date
No.
福
Have a nice ti
n2100
npc 10.
X!
(3) In the same Poisson process.
瑕疵
X = # of changes in the interval with length w.
Y =
"
If X~ PLA), then Y~ P (Kλ)
eg: Flows on a used computer tape occured on the
kw
Ch3
3-
process,
, what's the dist. of x,
average of I flaw pew 1200 feet.
In the
approx. Poisson
# of flaws in a 4800-foot rall ? X~P(4)
@ P(X ≤ 2) =
0
=0.238
✗!
Ⓒ p(x = 5) = = = P(xss)-P(x≤4) = 0/85-0.639
E(x)=4
=
eg: In a large city, telephoncalls to 911 come on.
the
average of 2 every
3 minutes in a
Poisson
process.
What's the prob. of 5 or more calls
of 5 or more calls arriving in a 9-min period?
X= # of calls in 9-min period.
X~P16)
P(x75) = (- p(x≤ 4) = (-0,285.
Have a nice time.