Summary Mathematical Statistics 214 Part 1

MathsStats Summary


ChapterFour



Continues Variable i take on any value in a given interval

Probability densityfunction


pay

A
Discrete Variable I Take on finite number of Values

Probability massfunction


pod

x


Cummulative DistributionFunction
stepfunction
Discrete


y PYay for Ryan
CDF

Properties FC o o continuous s Fal
Flo I
FLY isnondecreasing

, Probability density function continuesvariable I

PDF



fly
PDF
ddy
Fly Fly
co
2
Fly
g
f flyldy



ofPDF ffflyldy
Property PDFdoesnot havetobe continuous



Quartiles of distribution s

Thepthquantile of Y P isthesmallest valuesuchthat



Forcontinuous variable s PCYedp FyOp P



Theorem
i b
Playab fafeyldy




ExpectedValues




Discrete variable ELY dyPly
PDF

continuous variable ELY f yflyldy

,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