Sampling Distributions and Statistical Inference: A Basic Introduction, Study notes of Statistics

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* Parameters of a^ distribution are denoted as O

e.g.^ O^ for^

normal distribution →^ CM, 64

fix ;D^ )^ denotes^ pflpdf of^ a^ distribution^ with^ parameter 0

FIX :O)^ denotes^ Cdf

D

Simplest assumption { (^) Xi (^) , Xz^ ,^ Xs^ ' - ' Xn I are^ independent and^ identically distributed random variables

from distribution^ fix:O)

A random^ sample of size^ n from^ distribution^ fix :o) 6.c.IJoihdistribu-htmefarandomsample-i.ee

.^ Likelihood^ function By independence

The joint pflpdf B^ :

fix.^ ,Xa,

• -^ - . Xn) =

fcxi :O,^ fix

.io/---fCXnioI=I7fCxiiO16.5.lSamphhgoh3tributionofasta-h3t 0 A^ simple random sample

is modelled^ as^ a^

sequence of^ IID (^) random Variables. Sampling distribution is the^ probability distribution (^) of the values which the statistic would^ have^ in^ a large number^ of^ samples collected^ Cirdepardently) from the^ same population. ① French^ -^ E) (h

• y g^2

② g- n^ x'^ n - I

0 Size^ of^ n

For (^) symmetric distributions^ ,^ small^ n is enough For very skewed distribution (^). larger n^ B required n >^30 is^ generally sufficient (^) for a reasonable^ approximation 6.8GmmonSamphhgDBtributim D X ' distribution Let Zi (^) , Zz ,

• (^) " , Zk^ be^ independent Nco^ ,^ l^ )^ random^ variables^. :::i÷¥÷÷÷::::÷÷

Continuous for values of X 30

Zo

'

n XT for all i -_ 1,2, - - ' n

F- (^) Kkk

Varix )^ =^ 2K

¥÷i÷÷÷i÷:.:÷:::::nwa.*nx

0 Students

'

t distribution

÷÷÷÷÷÷÷::::÷:::

T distribution is^

symmetric

around (^0).

As k→x^ , th^ →^ Nco^ , I^ )

For finite^ values^ of^

k

, the^ distributions^ have^ heavier^ tails^ than^ the

standard normal distribution^.

For Tr ta

ECT) =^ O^ for k>^ I Var (^) CT) = ¥

for k^ >^2

O (^) F distribution "

÷÷÷÷÷÷÷:÷::÷÷÷:÷

Continuous (^) for Values of^ X > o

t