DISTRIBUTIONS OF FUNCTIONS OF

CONTINUOUS RANDOM VARIABLES

DISTRIBUTIONS OF DISTRIBUTIONS OF FUNCTIONS OF FUNCTIONS OF

CONTINUOUS RANDOM CONTINUOUS RANDOM VARIABLESVARIABLES

Normal, Gamma, Normal, Gamma, ExponensialExponensial, Chi, Chi--Square, Square, Student and FStudent and F

Normal Distribution• The important distribution in statistics

was found by DeMoivre (1733) and Gauss• Depend on two parameters : µ (population

mean) and σ (population standard deviation)

• Pdf for random variable normal X : n(x; µ, σ) or

xexf x ;2

1)( 2/)2/1(

Normal Curve

µ x• Normal distribution with µ=0 and σ=1 is

named Standard Normal Distribution

The characteristics of Normal Curve

1. Mode : x = µ2. Curve normal is symetri to mean µ3. Curve has ‘titik belok’ on :

x = µ ± σ, ‘cekung ke bawah’ if µ-σ<X<µ+σand ‘cekung ke atas’ for the others x

4. Y = 0 is an asymtoth for curve normal5. The area below this curve is 1

The area below of Normal Curve

• The area below of curve normal, between x=x1and x=x2 :

• Probability in one point x = c for continur.v.

2

1

2/)()2/1(21 2

1x

x

x dxexXxP

)()(such that 0)(

2121 xXxPxXxPaXP

x1 µ x2 x

)( areablack The 21 xXxP

Standardized• Given r.v. X~ N(µ,σ2)• Transformation :

make Z ~ N(0,1)

x1 x2 µ≠0 σ ≠1 z1 z2 µ=0 σ =1

σµXZ

Example 1 • Given X has normal distribution with µ=50 and σ=10. Count probability which X has values : between 45 and 62.

• Solution :

5764,03085,08849,0)5,0()2,1()2,15,0(

)10

506210

5045()6245(

ZPZPZP

ZPXP

Example 2• Suatu jenis baterai mobil mean berumur 3 tahun

with standard deviation 0,5 tahun. Bila umurbaterai berNormal distribution, berapa persenbaterai jenis A akan berumur kurang dari 2,3 tahun.

• Solution : Misal X : umur baterai mobil jenis A

= 8,08 %

2,3 3 x

0808,0)4,1()3,2(

ZPXP

The Central Limit TheoremGiven X has particular distribution with mean µ and standard deviation σ . If the sample (n) is big enough (n), then Z = (X- µ)/ σ has standard normal distribution N(0,1). ‘Limiting Distribution’Special cases : application of this theoremTheorem :

)1,0(~/

)/,(~ 2 Nn

XZnXn

Gamma Distribution • Gamma distribution gets the name from gamma

function :

• Pdf for continue random variable X, which has gamma distribution gamma with parameter α>0and β>0 :

• µ = α.β and σ2 = α.β2

0

1 0untuk ; 1)!-( )( dxex x

others , 0

0,)(

1)(

/1 xexxf

x

Exponensial Distribution• Pdf for continue random variable X which

has exponensial distribution with parameter β>0 :

• µ = β and σ2 = β2

others , 0

0 , 1)(

/ xexf

x

Chi-Square Distribution • Pdf for continue random variable X which

has chi-square disribution with degree of freedon (d.f.) ν :

• µ = ν and σ =2 ν with ν ‘bil. bulat +’;Chi-Square is gamma with α = ν/2 and β= 2.

lainnya untuk , 0

0 , )2/(2

1)(

2/12/2/

x

xexxf

x

The characteristics of Curve Chi-Square

1. The chi-square curve is in kwadran I 2. The curve is not symetri, has

tendency to the right (positifcurve).

3. Y = 0 is an asymptoth for this curve4. The area below the curve : 15. …

Curve Chi-Square

• The black area = p • Critical point for p=0,95 and ν = 14

is 23,7

2p

TheoremIf S2 is sample variantion which is come from normal population with variance σ2 , then random variable :

has chi-square distribution with d.f.: ν = n-1.

22

2

~)1(

Sn

Example 1• Search the critical point for df=9, if

the right area = 0,05 and the left area= 0,025 !

from table Chi-Square

=2,70=16,9

21

22

2122

Student Distribution • Almost rare population variance is known• For sample with n 30, good estimation

for σ2 is S2 or • If n < 30 we have t distribution

21nS

nSXT //

Student Distribution • W.S. Gosset who has found this

distribution first time in 1908

• His research was declared with name : “Student”

T distribution with d.f. : ν=n-1

Let r.v. standard normal and r.v. chi-square with

d.f. ν=n-1.If Z and V is independentt, then distribution of r.v. :

is given by :

)1/(/)1(

//22

nSnnXT

nXZ

/

2

2)1(

SnV

tvt

vvvtf

v

;1.2/

2/1)(2/12

Relation t curve with ν = 2 and 5 and standard normal with ν =

ν =

ν = 5ν = 2

0

The characteristics of Curve t

1. The t curve is symetri to mean = 02. The t curve shape like a bell, but t

distribution is different from Z because of the t’s value depend on two statistic : and S2 , Z’s value depend on

3. Y = 0 is an asymptoth for t’s curveasimtot datarnya

4. The area below the curve is 15. …

X X

F Distribution Let U and V are two dependent random variables which have chi-square with df1= ν1 and df2= ν2 ,then distribution of r.v :

with dk1= ν1 and dk2= ν2 FVUX ~

//

2

1

others , 0

0;/1

.2/2/

/.2/)( 2/

21

12

21

2/2121

21

11

xvxv

xvv

vvvvxf vv

vv

F Curve

F

• for p=0,05 with (ν1,ν2)=(24,8) : F=3,12for p=0,01 with (ν1,ν2)=(24,8) : F=5,28

),(;),();1(

21

12

1

pp F

F

The characteristics of F Curve

1. The curve is in kwadran I 2. The curve is not symetri, has tendency to

the right (positive curve)3. Y=0 is an asymptoth for this curve4. The area below the curve is 15. …• Sampling Distribution