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BAB I

DESCRETE DISTRIBUTION

A. DISCRETE UNIFORM

f(x) =

Mean () =

= E(x) =

N

1x

f(x) . x

=

E(x) = (terbukti)

Varians = 2 =

Bukti:

2 = E(x2) – (E(x))2

= E(x(x - 1)) + E(x) – (E(x))2

E(x(x - 1)) =

=

=

= [(12 + 22 + … + N2)] – [(1 + 2 + … + N)]

=

=

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=

=

E(x2) = E(x(x – 1)) + E(x)

= +

=

Var (x) = 2 = E(x2) – (E(x))2

=

=

Var (x) = 2 = (terbukti)

1. Moments

3 =

Bukti

3 = E(x3) =

=

=

=

=

=



3 = (terbukti)

4 =

Bukti

4 = E(x4) =

=

N

1x

4

N

1 . x

=

=

=

4 = (terbukti)

2. Moment Generating Function (MGF)

MGF = Mx(t) = E(etx) = tx

N

1x

e.N

1

Bukti

Mx(t) = )x(f.ex

tx

=

= tx

N

1x

e.N

1

Mx(t) = E(etx) = tx

N

1x

e.N

1

(terbukti)

B. Bernoulli

f(x) = px q1-x



1. Mean () = E(x) = p

= E(x) = N

x

f(x) . x

=

=

1-n

0x

1)-(x-1)-(11-x p)-(1 . p . p . x

= p .

= p .

Misal: m = n – 1

y = x – 1

E(x) = p .

= p . 1

= p

() = E(x) = p (terbukti)

2. Varians (2) = pq

Bukti

2 = E(x2) – (E(x))2

= p – p2

= p (1 – p)

= p.q

Varians (2) = p.q (terbukti)



3. Moments : r = p for all r

Bukti:

r = E(x) =

=

=

= p . dimana

= p . 1 = p

r = p for all r (terbukti)

4. MGF : Mx(t) = q + p . e1

Bukti:

Mx(t) = )x(f.ex

tx

=

=

= q + p . et

Mx(t) = q + p . e1 (terbukti)



C. BINOMIAL

f(x) = px qn-x

1. Mean = n . p

Bukti:

E(x) = x

f(x) . x

=

= )1x()1n(1x

1n

1x

)p1(pp))!1n()1n(()!1x(x

)!1n(nx

= n . p .

Misal: m = n – 1

y = x – 1 untuk x = 1, y = 0

2. Varians

Var (x) = (2) = n . p . q

Bukti:

Var(x) = E(x2) – (E(x))2

E(x2) = E(x(x – 1)) + E(x)

E(x(x – 1)) =

=

=

= n p (n – 1)

Misalnya: m = n – 2

Y = x – 2 untuk x = 2, y = 0



E(x(x – 1)) = n2 p2 – np2 dimana

m

0y

ymy 1qp)!ym(!y

!m

= n2 p2 – np2 . 1

E(x2) = E(x(x – 1)) + E(x)

= n2 p2 – np2 + np – n2 p2

Var(x) = E(x2) – (E(x))2

= n2 p2 – np2 + np – n2 p2

= - np2 + np = np – np2 = np (1 – p)

= 2 = npq

Var(x) = 2 = npq (terbukti)

3. MGF

Bukti:

Mx(t) = E(ext) = (q + p et)n ; menggunakan moment generating function

(MGF)

f (x) = ; x = 0, 1, 2, 3, … n

Mx(t) = E(ext) =

=

=

= (q + p . et)n

Jadi terbukti Mx(t) = E(ext) = (q + p et)n



D. HIPER-GEOMETRIK

f(x) = ; x = 0, 1, 2, ..

1. Mean

Bukti: Mean menggunakan definisi mean

N .

E(x) =

=

=

=

= k

= k

Misalkan: y = x -1 x = y + 1

X = 1 y = 0

= k

= k

= k

= n . . 1

= n .



Jadi terbukti Mean = n .

2. Varians

Bukti:

Var(x) = n . ; menggunakan definisi mean.

Var(x) = E(x2) – [E(x)]2

E(x2) = E(x2) – E(x) + E(x)

= E[x(x – 1] + E(x)

E[x(x – 1)] =

=

=

=

=

= k(k – 1)

= k(k – 1)


X = 2 y = 0

E[x(x – 1)] = k(k – 1)

= k(k -1)

= k(k – 1)



= n

E(x2) = E[x(x -1)] + E(x)

=

Var(x) = E(x2) – [E(x)]2

= n .

=

Jadi terbukti Var (x) =

3. MGF not useful.

E. DISTRIBUSI POISSON

f(x) = !x

.e x

; x = 0, 1, 2, ..

1. Mean

Bukti: Mean menggunakan definisi mean

Mean = E(x)

=

=

=

=

Misalkan: y = x -1

x = 1 y = 0

=

= . 1

=



Jadi terbukti Mean =

2. Varians

Bukti: Var(x) menggunakan definisi mean.

Var(x) = E(x2) – [E(x)]2

E(x2) = E(x2) – E(x) + E(x)

= E[x(x – 1] + E(x)

E[x(x – 1)] =

=

=

= 2


X = 2 y = 0

E[x(x – 1)] = 2

E(x2) = E[x(x -1)] + E(x)

= 2 +

Var(x) = E(x2) – [E(x)]2

= 2 + - 2

=


3. Moment

Bukti:

E(xr) = r

=

= e-



= e- .

= e- e =

Terbukti E(xr) =

4. MGF

Bukti:

f(x) = !x

.e x

; x = 0, 1, 2, ..

Mx(t) = E(ext) =

= e-

= e- e.et

= exp. [ (et – 1)]

Jadi terbukti Mx(t) = E(ext) = exp. [ (et – 1)]

F. GEOMETRIC

f(x) = p . qx - 1 ; x = 0, 1, 2, ..

1. Mean

Bukti: Mean = menggunakan definisi mean

Mean = E(x)

=

=

=



=

=

= ; dimana = 1

= . 1 =


2. Varians

Bukti: Var(x) = menggunakan definisi mean.

Var(x) = E(x2) – [E(x)]2

E(x2) = E(x2) – E(x) + E(x)

= E[x(x – 1] + E(x)

E[x(x – 1)] =

=

Var(x) = E(x2) – [E(x)]2

E(x2) =

=

=

= q(q + 1)

=



=

Var(x) = E(x2) – [E(x)]2

= -

=


3. MGF

Bukti:

Mx(t) = E(ext)

f(x) = p . qx ; x = 0, 1, 2, 3, ..

Mx(t) = E(ext) =

= p

= p =

Jadi terbukti Mx(t) = E(ext) =

G. NEGATIVE BINOMIAL

f(x) = pr . qx ; x = 1, 2, 3, ..

1. Bukti : Mean menggunakan definisi mean

Mean = E(x)

=

=

=



=

=

=

=

= r . q . 1 + q . x . 1

E(x) = r . q + q E(x)

E(x)-q.E(x) = r . q

E(x) (1 – q) = r . q

E(x) = ; 1 – q = p

=


2. Bukti: Var(x) = menggunakan definisi mean.

Var(x) = E(x2) – [E(x)]2

E(x2) = E(x2) – E(x) + E(x) = E[x(x – 1] + E(x)

E[x(x – 1)] =

=

=

=

= q2



= q2

= q2 + q2

= r . q2 + q2

Var(x) = E(x2) – [E(x)]2

= q2 (r + 1) -

= q2 (r + 1) -

=


3. Bukti: Mx(t) = E(ext) ; menggunakan moment generating function (MGF)

f(x) = pr . qx ; x = 1, 2, 3, ..

Mx(t) = E(ext) =

= pr

= pr

=

Jadi terbukti Mx(t) = E(ext) =



BAB 2

CONTINUES DISTRIBUTION

A. UNIFORM RECTANGULAR

f(x) =

1. Mean

Mean : () = E(x) =

=

Terbukti = E(x) =

2. Variance

Var(x) = 2 =

Bukti:

E(x2) =

Var(x) = E(x2) – [E(x)]2

=

=

Var(x) =

B. EKSPONENSIAL

1. Mean

Bukti :



Misalkan : u = x du = dx

dv = -λ . e-λ.x dx v = -e-λ.x


2. Varians

Bukti : menggunakan definisi mean.

Var(x) = E(x2) – [E(x)]2

Misalkan : u = x2 du = 2x . dx

du = -λ. e-λ.x dx v = -e-λ.x



Jadi terbukti Var(x) =

3. Moment

4. MGF

Bukti : Mx(t) = E(xxt) ; menggunakan moment generating function (MGF)

f(x) = λ ext ; x[0,]

Misalkan : y = (t – λ)x

Jadi terbukti Mx(t) = E(ext)

C. DISTRIBUSI NORMAL

1. Mean = = E(x)



Bukti :

E(x) = (Terbukti)

2. Variance

Var (x) = 2

Bukti : Var(x) = E(x2) – (E(x))2

Bukti :

Var (x) = E(x2) – (E(x))2

= 2 + 2 - 2 = 2

Var (x) = 2 (Terbukti)

3. MGF

*) Bukti : ; menggunakan moment

generating function (MGF)



Misalkan : w = [x – ( + t 2)]/

D. GAMMA

1. Mean

Bukti : Mean = menggunakan definisi mean.

Mean = E(x)



Misalkan : t = x

Jadi terbukti =

2. Varians

Bukti : Var(x) menggunakan definisi mean.

Var(x) = E(x2) – [E(x)]2

Misalkan :



Var(x) = E(x2) – [E(x)]2

Jadi terbukti Var(x)

3. Moment

(Terbukti)

4. MGF

Bukti : Mx(t) = E(ext) ; menggunakan moment generating function (MGF)

; x[0,]



Misalkan : y = (t - )x dy = (t - ). dx

Jadi terbukti

E. CAUCHY

1. Mean

Bukti :

F. DOUBLE EXPONENTIAL

1. Mean



2. Variance

3. MGF

BAB III

CONTINUES DISTRIBUTION (CONTINUED)

A. DISTRIBUTION WEIBUL

f(x) = abxb-1 ; x = 0, 1, 2, 3, .....

1. Mean

Bukti mean :

Mean



Misalnya :

2. MGF

Bukti :

Misalkan y = -a.xb



Jadi terbukti

B. LOGISTIC

1. Mean



2. Variance

C. PARETO

Bukti :

1. Mean



2. Variance



3. Moment

Misal



D. DISTRIBUSI t

dimana k > 0

a. Mean

Bukti : E(x) = 0

Misal :

b. Variance

Bukti : k/k -2

C. DISTRIBUSI BETA

a. Mean

Bukti Mean



b. Variance

Bukti :

Var (x) = E(x2) – [E(x)]2

Var (x)

Terbukti Var (x)

c. Moments

Bukti :

Terbukti E(xr)



DISTRIBUSI DISCRETE UNIFORM

*). Bukti : Mx(t) = E( ext ) = ; menggunakan moment


f(x) =

Mx(t) = E( ext) =



=

=

Jadi terbukti Mx(t) = E( ext) =

DISTRIBUSI DISCRETE BERNOULLI

*). Bukti : Mx(t) = E( ext ) = q + p.et ; menggunakan moment generating function (MGF)

f(x) =

Mx(t) = E( ext) =

=

=

= ( e t . p)O . q 1 + ( e t . p ) 1 . q O = 1. q + (e t . p) .1= q + p.e t

Mx(t) = E( ext) = q + p.e t ( terbukti )

DISTRIBUSI BINOMIAL NEGATIF

f(x) =

a. Bukti : Mean = menggunakan definisi mean.

Mean = E(x)

=

=

=

=



=

=

=

=

= r.q.1 + q.x.1E(x) = r.q + q E(x)

E(x)-q.E(x) = r.qE(x)(1-q) = r.q

E(x) = ; 1 – q = p

=


b. Bukti : Var(x) = menggunakan definisi mean.

Var(x) = E(x2) - [E(x)]2

E(x2) = E(x2) - E(x) + E(x) = E[x(x-1)] + E(x)

E[x(x-1)] =

=

=

=

=

=

= +



=Var(x) = E(x2) - [E(x)]2

=

=

=


c) Bukti : Mx(t) = E( ext ) ; menggunakan moment generating function (MGF)

f(x) =

Mx(t) = E( ext) =

=

=

=


DISTRIBUSI EKSPONENSIAL f(x) =


Mean = E(x)

=

=

Misalkan : u = xdu = dxdv = -. e -.x dxv = -e -.x

=



=

=

=



Var(x) = E(x2) - [E(x)]2

E(x2) =

=

Misalkan : u = x2 du = 2x. dxdv = -. e -.x dxv = - e -. x

=

=

=

Var(x) = E(x2) - [E(x)]2

=

=


d) Bukti : Mx(t) = E( ext ) ; menggunakan moment generating function (MGF)

f(x) =

Mx(t) = E( ext) =

=

=

Misalkan : y = (t-)xdy = t -. dx



dx = dy

Mx(t) =

=

=

=

=


DISTRIBUSI POISSON.

f(x) =

a. Bukti : Mean menggunakan definisi mean.Mean = E(x)

=

=

=

=

Misalkan : y=x-1x=1 y=0

=

= . 1=

Jadi terbukti Mean = b. Bukti : Var(x) menggunakan definisi mean.

Var(x) = E(x2) - [E(x)]2

E(x2) = E(x2) - E(x) + E(x)= E [ x ( x – 1 ) ] + E(x)



E [x(x –1)] =

=

=

=

Misalkan : y=x-2 x=y+2x=2 y=0

E [x(x –1)] =

E(x2) = E [ x ( x – 1 ) ] + E(x)= 2 +

Var(x) = E(x2) - [E(x)]2

= 2 + - 2 =

Jadi terbukti Var(x) = e) Bukti : menggunakan moment generating function (MGF)

f(x) =

Mx(t) = E( ext) =

=

=

== exp. [ (e t – 1 ) ]

Jadi terbukti Mx(t) = E( ext) = exp. [ (e t – 1 ) ]

DISTRIBUSI WEIBUL



f(x) =

Mx(t) = E( ext) =



=

Misalkan:

=

=

=

=


DISTRIBUSI CHI SQUERE



f(x) =

Mx(t) = E( ext) =

=

=

=



=

=

=

=

=

Mx(t) = E( ext) = ( terbukti )

DISTRIBUSI BINOMIAL (diskrit)*). Bukti : Mx(t) = E( ext ) = ; menggunakan moment


f(x) =

Mx(t) = E( ext) =

=

=

==


DISTRIBUSI NORMAL (continuous)





f(x) =

Mx(t) = E( ext) =

=

=

=

=

=

Misalkan : w = [x – ( + t 2 )] /

dw =

dx = dw

=

=

=

Mx(t) = E( ext) = ( terbukti )

DISTRIBUSI GAMMA

f(x) =


Mean = E(x)

=

=

=

Misalkan : t = x



x =

dt = dx

dx =

=

=

=

=

=



Var(x) = E(x2) - [E(x)]2

E(x2) =

=

=

Misalkan : t = x

x =

dt = dx

dx = dt

=



=

=

=

=

=

=

Var(x) = E(x2) - [E(x)]2

= - =

=


c. Bukti : Mx(t) = E( ext ) ; menggunakan moment generating function (MGF)

f(x) =

Mx(t) = E( ext) =

=

=

Misalkan : y = (t-)xdy =( t -). dx

dx = dy



x =

Mx(t) =

=

=

=

=

=


DISTRIBUSI GEOMETRI.f(x) =


Mean = E(x)

=

=

=

=

=

= ;

=





Var(x) = E(x2) - [E(x)]2

E(x2) =

=

=

=

=

=

=

Var(x) = E(x2) - [E(x)]2

=

=


d. Bukti : Mx(t) = E( ext ) ; menggunakan moment generating function (MGF)

f(x) =

Mx(t) = E( ext) =

=

=

=

=




DISTRIBUSI HIPER-GEOMETRIK.

f(x) =

a. Bukti : Mean menggunakan definisi mean.

Mean = E(x) =

=

=

=

=

=

=

Misalkan : y=x-1 x=y+1x=1 y=0

=



=

=

= . 1

=


b. Bukti : Var(x) = ; menggunakan definisi mean.

Var(x) = E(x2) - [E(x)]2

E(x2) = E(x2) - E(x) + E(x)= E [ x ( x – 1 ) ] + E(x)

E [x(x –1)] =

=

=

=

=



=

Misalkan : y = x-2 x = y+2x = 2 y = 0

E [x(x –1)] =

=

=

=

E(x2) = E [ x ( x – 1 ) ] + E(x)

=

Var(x) = E(x2) - [E(x)]2

=

=

Jadi terbukti Var(x) = E(x2) - [E(x)]2


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