Samenvatting (Examen)Formularium Statistiek Voor Psychologen Deel 2 (P0M17A)
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Course
Statistiek Voor Psychologen Deel 2 (P0M17A)
Institution
Katholieke Universiteit Leuven (KU Leuven)
Formularium te gebruiken op het examen Statistiek voor Psychologen deel 2. Document is een word document zodat deze nog aangepast kan worden naar eigen wensen en eigen toevoegingen. Formularium bevat ook stappenplannen van onder andere hypothesetoetsing en betrouwbaarheidsinterval. Ook formules van...
1. Modellen: Aantal keer succes of successen Wacht”tijd” tot eerste succes X Bern ( θ ) D ( 0<θ< 1 )
π X ( 1 )=P ( X =1 )=θ
Beurten of n Y Bin ( n , θ ) D ( 0<θ<1 ) Z Geo (θ ) D ( 0<θ <1 )
herhalingen π X ( 0 ) =P ( X=0 )=1−θ
()
π Y ( k )= n θ k ¿
k
π Z ( k ) =¿
1 2 1−θ
μZ = σ Z = 2
2
μ X =θ σ X =θ ( 1−θ )
2
μY =nθ σ Y =n(θ) ( 1−θ ) θ θ Bern (θ )=Bin(1 , θ)
Continu medium X Poisson ( λ ) D ( λ> 0 ) T exp ( λ ) C ( λ>0 ) Y
Y Bin ( n , θ ) voor proportie
{ n
k
λ −λ −λt
φ T ( t ) = λ e t ≥0
π X ( k )= e
k!
2
μ X =λ σ X = λ 1
μT = σ 2X = 2
0t <0
1
μ Y =E
n
Y
n [ ]
1
= E [ Y ] =θ
n
λ λ 1 θ ( 1−θ )
σ 2Y = 2 σ 2Y =
ФT ( t )=P ( T ≤ t )=1−e
−λt
n n n
(als bin geldt voor aantal succes Y in n
herhalingen van een bern-experiment)
X U ( a , b ) C( a<b) X N ( μ , σ 2 ) C(σ >0)
{
1
φ X ( x )= b−a voor a ≤ x ≤ b
2 ( σ )
2
−1 x−μ
1
¿ 0 anders φ X ( x )= e Strikt dalende functie van
√2 π ⋅ σ
a+b 2
μX= σ =¿ ¿ ( x−μ )2
2 X
d−c Dit impliceert dat φ X symmetrisch is t.o.v. μ
Als X U ( a , b ) en [ c , d ] ⊂ [ a , b ] dan P ( c ≤ X ≤ d )= 2 2
b−a μ X =μ σ X =σ
ZRM: a+(b−a)∗rand Standaardnormaalmodel: μ X =0 σ X =1
2
Als X N ( μ , σ ) en Y =aX+ b , danY N ( aμ+ b , a σ )
2 2 2
Modellen voor meerdere variabelen:
Biv. Onafhankelijk: π X , Y ( x , y ) =π X ( x ) ∙ π Y ( y )
2 ( σ[ ) ( )]
2 2
−1 x−μ1 y−μ2
+
1 σ2
π X ∨Y = y =π X ( x ) en π Y ∨X =x =π Y ( y ) N : φ X ,Y ( x , y )= e
1
2 π ⋅σ 1 σ 2
φ X ∨Y = y =φ X ( x ) en φ Y ∨X =x =φ y ( y )
Bern: π X ,Y ( 0,0 )=(1−θ1 )(1−θ 2)
Biv. Afhankelijk:
π x , y (x , y ) = π X ∨Y = y (x )∙ π Y ( y ) N :φ X ,Y ( x , y )=
1
e
−1
2( 1−ρ )
2 [( ) ( )
x−μ1 2 x−μ2 2 2 ρ ( x−μ1 )( y−μ2 )
σ1
+
σ2
−
σ 1 ∙σ 2
2 π ⋅σ 1 σ 2 ( 1−ρ )
2
= π Y ∨X = x ( y)∙ π X ( x)
2 2
Te noteren als: ( X , Y ) N ( μ1 , μ2 ; σ 1 , σ 2 , ρ)
Complexe modellen:
Mengselmodellen: π x = λ ∙ π (1) (2 )
x +(1−λ)∙ π x
π x = λ 1 ∙ π (x1) + λ2 ∙ π (x2) +(1−λ1− λ2) ∙ π (3)
x
Regressiemodellen: 2
Y ¿ X =x N (β 0 + β 1 x j , σ )
j
2
Y i=β 0 + β 1 x i + Ei met Ei (iid ) N ( 0 , σ )
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