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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