Distributions

Distributions∗ USC Linguistics August 17, 2007

binomal pdf:  p(c(t)) = 

n k

 =

N c(t)



p(t)c(t) (1 − p(t))N −c(t)

(1)

n! n ∗ (n − 1) ∗ ... ∗ (n − (k − 1)) = k!(n − k)! k!

(2)

 N  X N p(t)ci (t) (1 − p(t))N −ci (t) p(c (t) > ci (t)) = ci (t) 0

(3)

i

Binomial PDF; p(t)=.5 0.18 0.16 0.14 0.12 p(c(t))

0.1 0.08 0.06 0.04 0.02 0 0

5

10 c(t); N=20 events

15

20

multinomial pdf: p(c(v1 ), ..., c(vk )) =

N! p(v1 )c(v1 ) ...p(vk )c(vk ) c(v1 )!...c(vk )!

∗

(4)

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normal distribution:

(x−µ)2 1 (− √ e 2σ2 ) σ 2π

(5)

Normal PDF 0.18 µ = 10, σ =

0.16

√

5

0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0

5

10

15

20

µ = N p : 20 ∗ .5 = 10

(6)

σ 2 = N p(1 − p) : 10 ∗ .5 = 5

(7)

e

Z ∞ X 1 1 e= ; dx = 1 k! x k=0

(8)

1

1 δ logb e δ ; loge (x) = logb (x) = x δx x δx

(9)

poisson distribution:

e−E(c(t))E(c(t))c(t) p(c(t)) = c(t)! E(c(t)) =

totaltime 10minutes = = 2.5 averagetime/event 4minutes/event E(c(t)) = µ = σ 2

2

(10) (11) (12)

Normal & Poisson PDFs 0.14

√ µ = 10, σ = 10 E(c(t)) = 10

♦ ♦

0.12

♦

♦

0.1 0.08

♦

♦

0.06

♦

0.04

♦

0.02 0♦ ♦ ♦ 0

♦

♦

p(c(t))

♦

♦ ♦

♦ ♦ 15

10 c(t)

5

gamma function:

Z∞ Γ(α) =

xα−1 e−x dx

♦

♦ ♦ ♦ 20

(13)

0

for α ∈ N: Γ(α) = (α − 1)! gamma pdf: p(x) =

λα α−1 −λx x e Γ(α)

χ2 : f◦ =degrees of freedom α= p(x) =

(14)

f◦ 1 ,λ = 2 2

( 21 )

f◦ 2

◦ Γ( f2 )

x

f◦ −1 2

(16) 1

e− 2 x

f ◦ = (#rows − 1)(#columns − 1) E=

T otal(row) ∗ T otal(col) N χ2 =

X (O − E)2 E

3

(15)

(17) (18) (19)

(20)

Gamma PDF (χ2 : f ◦ = 2α; λ = .5) 0.5

α = 10, λ = 2 α = 5, λ = 2 α = 3, λ = .5 α = 1, λ = .5 α = .5, λ = .5

0.4 0.3 0.2 0.1 0 0

2

4

E[X] =

6

8

10

α α ; V ar[X] = 2 λ λ

(21)

E[χ2 ] = 2α = f ◦ ; V ar[χ2 ] = 4α = 2f ◦ Observed poor rich greedy 80 40 lazy 60 20 total 140 60 χ2 =

Expected poor rich greedy 84 36 lazy 56 24 total 140 60

120 80 200

16 16 16 16 + + + = 1.59 84 36 56 24

p(χ2 > 1.59, f ◦ = 1) = .2073

(22)

120 80 200 (23) (24)

References Baird, Davis (1983) “The Fisher/Pearson Chi-Squared Controversy: A Turning Point for Inductive Inference,” Br J Philos Sci, 34(2), 105–118.

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