Probability and Distributions. A Brief Introduction

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1 Probability ad Distributios A Brief Itroductio

2 Radom Variables Radom Variable (RV): A umeric outcome that results from a experimet For each elemet of a experimet s sample space, the radom variable ca take o exactly oe value Discrete Radom Variable: A RV that ca take o oly a fiite or coutably ifiite set of outcomes Cotiuous Radom Variable: A RV that ca take o ay value alog a cotiuum (but may be reported discretely ) Radom Variables are deoted by upper case letters (Y) Idividual outcomes for RV are deoted by lower case letters (y)

3 Probability Distributios Probability Distributio: Table, Graph, or Formula that describes values a radom variable ca take o, ad its correspodig probability (discrete RV) or desity (cotiuous RV) Discrete Probability Distributio: Assigs probabilities (masses) to the idividual outcomes Cotiuous Probability Distributio: Assigs desity at idividual poits, probability of rages ca be obtaied by itegratig desity fuctio Discrete Probabilities deoted by: p(y) = P(Y=y) Cotiuous Desities deoted by: f(y) Cumulative Distributio Fuctio: F(y) = P(Y y)

4 Discrete Probability Distributios Probability (Mass) Fuctio: p( y) P( Y y) p( y) 0 y all y py ( ) 1 Cumulative Distributio Fuctio (CDF): F( y) P( Y y) F( b) P( Y b) p( y) F( ) 0 F( ) 1 F( y) is mootoically icreasig i y b y

5 Cotiuous Radom Variables ad Probability Distributios Radom Variable: Y Cumulative Distributio Fuctio (CDF): F(y)=P(Y y) Probability Desity Fuctio (pdf): f(y)=df(y)/dy Rules goverig cotiuous distributios: f(y) 0 y f ( y) dy 1 P(a Y b) = F(b)-F(a) = b f ( y ) dy a P(Y=a) = 0 a

6 Expected Values of Cotiuous RVs Expected Value: E( Y ) yf ( y) dy (assumig absolute covergece) E g( Y ) g( y) f ( y) dy Variace: ( ) ( ( )) ( ) ( ) V Y E Y E Y y f y dy y y f ( y) dy y f ( y) dy yf ( y) dy f ( y) dy E Y ( ) (1) E Y E ay b ( ay b) f ( y) dy a yf ( y) dy b f ( y) dy a( ) b(1) a b V ay b E ( ay b) E( ay b) ( ay b) ( a b) f ( y) dy ay b ( ay a) f ( y) dy a ( y ) f ( y) dy a V ( Y ) a a

7 Meas ad Variaces of Liear Fuctios of RVs i1 U a Y a costats Y radom variables i i i i, E Yi i V Yi i COV Yi Yj E Yi i Yj j ij E U E aiyi aii i1 i1 V U V a Y a a a 1 i i i i i jij i1 i1 i1 ji1 Y1,..., Y idepedet V U V a Y i i i 1 i1 a i i

8 Normal (Gaussia) Distributio Bell-shaped distributio with tedecy for idividuals to clump aroud the group media/mea Used to model may biological pheomea May estimators have approximate ormal samplig distributios (see Cetral Limit Theorem) Notatio: Y~N(, ) where is mea ad is variace 1 ( y) 1 f y e y ( ),, 0 Obtaiig Probabilities i EXCEL: To obtai: F(y)=P(Y y) Use Fuctio: =NORMDIST(y,,,1) Virtually all statistics textbooks give the cdf (or upper tail probabilities) for stadardized ormal radom variables: z=(y-)/ ~ N(0,1)

9 f(y) Normal Distributio Desity Fuctios (pdf) Normal Desities N(100,400) N(100,100) N(100,900) N(75,400) N(15,400) y

10 Secod Decimal Place of z Iteger part ad first decimal place of z 1-F(z)

11 Chi-Square Distributio Idexed by degrees of freedom () X~c Z~N(0,1) Z ~c 1 Assumig Idepedece: X,..., X ~ c i 1,..., X ~ 1 Desity Fuctio: i i c i i1 1 0, 0 1 x f x x e x Obtaiig Probabilities i EXCEL: To obtai: 1-F(x)=P(X x) Use Fuctio: =CHIDIST(x,) Virtually all statistics textbooks give upper tail cut-off values for commoly used upper (ad sometimes lower) tail probabilities

12 f(x^) Chi-Square Distributios Chi-Square Distributios df= df= df=0 df=30 df=50 f1(y) f(y) f3(y) f4(y) f5(y) X^

13 Critical Values for Chi-Square Distributios (Mea=, Variace=) df\f(x)

14 Studet s t-distributio Idexed by degrees of freedom () X~t Z~N(0,1), X~c Assumig Idepedece of Z ad X: T Desity Fuctio: f t Z ~ t X 1 1 t 1 t 0 Obtaiig Probabilities i EXCEL: To obtai: 1-F(t)=P(T t) Use Fuctio: =TDIST(t,) Virtually all statistics textbooks give upper tail cut-off values for commoly used upper tail probabilities

15 Desity t(3), t(11), t(4), Z Distributios f(t_3) f(t_11) f(t_4) Z~N(0,1) t (z)

16 Critical Values for Studet s t-distributios (Mea=, Variace=) df\f(t)

17 F-Distributio Idexed by degrees of freedom ( 1, ) W~F 1, X 1 ~c 1, X ~c Assumig Idepedece of X 1 ad X : W X X 1 1 ~ F, 1 Desity Fuctio: w f w w 1 w 0 1 1, 0 1 Obtaiig Probabilities i EXCEL: To obtai: 1-F(w)=P(W w) Use Fuctio: =FDIST(w, 1, ) Virtually all statistics textbooks give upper tail cut-off values for commoly used upper tail probabilities

18 Desity Fuctio of F F-Distributios f(5,5) f(5,10) f(10,0) F

19 Critical Values for F-distributios P(F Table Value) = 0.95 df\df

Joint Probability Distributions and Random Samples. Jointly Distributed Random Variables. Chapter { }

Joint Probability Distributions and Random Samples. Jointly Distributed Random Variables. Chapter { } UCLA STAT A Applied Probability & Statistics for Egieers Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistat: Neda Farziia, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig