These notes cover the first half of the syllabus for the Mathematical Statistics 214 module at Stellenbosch university. The notes include all theory from textbook and lectures as well as worked examples for each section.
,Function of a randomvariable s EEgly
f glylfoyldy
VLy and Ely
ECO C Vly EL ut
ly
EL cgey C EEgly
Vig ELY CELYN
EEgilgit grilyn ELgicy.pt tEIgneyn Vlaytb a var y
Uniform Distribution t
Y Unit0,02 E Y VCU andfell are onformulasheet
Normal Distribution 1
Y N M04 FLY VCU andfell are onformulasheet
Wecan alwaystransform a normal random variable to a standard normal
Variable usingthe following relationship
2 Y
,Gamma Distribution
Y gammalaB FLY VCU andfell are onformulasheet
Gammafunction a
f y
é dy
Chisquare Gamma Distribution
Y is chisquare if it is a gamma random
variable with a 2 and13 2
degrees of freedom
ELY VCU andfell are onformulasheet
Wecan convert a gamma variable to chi
square andthen usethetablesto find probability
Y gamma a B I 4 43 Kaa
Exponential Distribution
Y exp B FLY VCU andfell are onformulasheet
Beta Distribution s
FLY VCU andfell are onformulasheet
Y Beta QB
Bla 13 CH B
Betafunction 2 13
,Momentsandmoment functions
generating
TheKthnon centralmomentof a random variableY I firstnoncentral Mi M E Y
Mk ELY
The Kth central moment of a random variableY I first central ELYMI EY n o
My EL Y a I Second central Varcy
If two variables havethesamemomentsthen they havethe same PDFand Cbt
Momentgeneratingfunctions
Mct E et k Mt ECety
ety.pe
continuous met ECety JetYfogdy
Mdtmet to
Mk
Moment functionof a function of a variable
generating
If Yis a random variableand
galis a function of Ythen the moment generating function
for y is
g
Let.ge
f et.ge
fogdy
, TchebysheffsTheorem
Let Y be a random variable withfinitemean u and variance o then
for any K10
P Y M L KO I K2 OR PC Y MI KO f k
Gives probability that is ko unitsfrom mean true for distribution
y any
ChapterFive Bivariatedistributions
Distributions with two random variables
Discrete variables 1
equivalentto pmf
Joint bivariate probability function Ply ya PYagi Ya ya
Bivariate CDF
Fly ya Ey Ily PCtita
NB 1
Paige 421yd Flyya
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