STATISTICAL METHODS FOR BUSINESS

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1 STATISTICAL METHODS FOR BUSINESS UNIT 5. Joit aalysis ad limit theorems dimesio distributios. Margial ad coditioal distributios Sequeces of idepedet radom variables. Properties Sums of radom variables Cetral Limit Theorem ad its applicatios

2 UNIT 5. OBJECTIVES To apply the mai properties derived from idepedece of radom variables. To calculate probabilities for the mai aggregates of idepedet radom variables. To apply ad iterpret the Cetral Limit Theorem.

3 STATISTICAL METHODS FOR BUSINESS UNIT 5. Joit aalysis ad limit theorems dimesio distributios. Margial ad coditioal distributios.

5 Discrete bidimesioal radom variables Joit observatio of two discrete radom variables ad. (,) DISCRETE RVs {(x i,y j)/i, j = 1,2,...} P:(x,y) R 2 P(x,y) = P(=x, =y) [0,1] \ x 1... x i... x k y 1 p 11 p i1 p k1 Joit probability p ij =P(=x i,=y j )... P( x, i y j ) p ij y j p 1j p ij p kj... y l p 1l p il p kl i j p ij 0 p ij =1

6 Margial distributios \ x 1... x i... x k y 1 p 11 p i1 p k1... Margial distributio of : (y j, p.j ) y j p 1j p ij p kj... å p.j= pij i y l p 1l p il p kl å p i. = pij j Margial distributio of : (x i, p i. )

7 Margial distributios The margial distributios are uivariate distributios, so PROBABILITIES ad all the characteristics of ad may be computed i a straightforward way: E() Var() E() Var()

8 Cotiuous bidimesioal variables Joit observatio of two cotiuous Rvs, ad. (,) CONTINUOUS RVs Joit desity fuctio f:(x,y) R 2 f(x,y) R f(x, y) 0 Margial desity fuctios f (x,y )dxdy=1 f (y)= ò + - f(x,y)dx f ( x)= f (x,y )dy

9 Joit distributio fuctio For ay bidimesioal RV (,), its distributio fuctio is defied as follows: Joit distributio fuctio F:(x,y) R 2 F(x,y) = P(x, y) [0,1] [The value of the joit distributio fuctio at (x,y) is the total probability accumulated up to poit (x,y).]

10 Liear correlatio Covariace σ =Cov, =E éë -μ -μ ( ) ( )( ), ùû σ, =E -E E ( ) ( ) ( ) Liear correlatio coefficiet σ, ρ, =, σ σ 1 1 The sig of, idicates the type of correlatio (direct, iverse). The absolute value of, idicates the itesity of correlatio.

11 ( Vector Radom vectors. Characteristics Expectatio Variaces- Covariaces Uivariate μ=e( ) σ 2 =Var( ) Bivariate (,) -variate ( 1, 2,..., ) μ=( E ( ),E( )) μ=( E ( 1 ),..., E( ) ) 2 æσ ö σ ç 2 èσ σ ø 2 σ 1 σ σ 1 2 σ 21 σ 2... σ 2 ) σ 1 σ 2... σ

12 STATISTICAL METHODS FOR BUSINESS UNIT 5. Joit aalysis ad limit theorems Sequeces of idepedet radom variables. Properties

13 Idepedet radom variables Two RVs ad are idepedet iff: F(x,y) = F (x) F (y) (x,y) 2 Joit distributio fuctio of (,) Margial distributio fuctio of Margial distributio fuctio of INDEPENDENCE CONDITION (,) DISCRETE P(x i,y j )=P(x i )P(y j ) (x i,y j ) 2 (,) CONTINUOUS f(x,y)=f (x)f (y) (x,y) 2

14 Properties of fuctios of idepedet RVs Give ad, idepedet RVs, it holds: E() = E()E(). Cov(,) = 0 ad =0. Var(+) = Var() + Var(). Var(a+b) = a 2 Var() + b 2 Var(), a,bir. Var(-) = Var() + Var(). If ad are ucorrelated ormal RVs ( =0), the ad are idepedet

15 Reproductivity A class of RVs is reproductive if for every idepedet 1, 2 it holds Idepedet RVs Sum Reproductivity B( B(,,p) p) B(,p) The biomial model is reproductive i (for ay fixed p) N(μ N(μ,σ,σ ) ) N(μ μ, σ 2 σ 2 ) The ormal models is reproductive i mea ad variace P(λ P(λ ) ) P(λ λ ) The Poisso model is reproductive i

16 STATISTICAL METHODS FOR BUSINESS UNIT 5. Joit aalysis ad limit theorems Sums of radom variables

17 Sum of radom variables 1 (Profits of firm 1) 2 (Profits of firm 2)... S = Sum å i=1 Total profit of the firms i (Profits of firm ) = Mea å i=1 Mea profit of the firms i

18 Characteristics Give a -dimesio RV ( 1, 2,, ) with fiite expectatios ad variaces, amely: The followig holds: Expectatio ( ) ( ) ( ) ( ) ( ) ( ) E =μ E =μ... E =μ Var =σ Var =σ... Var =σ E (S )= i =1 μ i ( ) å i=1 E = μ i Variace Var (S )=Var( i=1 i )= i=1 σ i 2 + i j σ ij Var (S )= i=1 If 1, 2,..., are idepedet: 2 σ i Var ( )= i=1 σ i 2

19 Tchebyshev's Iequality Give ( 1, 2,, ), a -dimesio RV with fiite expectatios ad variaces, it holds: Tchebyshev's boud for the sum: P ( S E ( S N ) ε ) Var ( S ) ε 2 Tchebyshev's boud for the mea: P ( E ( ) ε ) Var ( ) ε 2

20 Idepedet idetically distributed RVs Give 1, 2,..., idepedet idetically distributed (iid) RVs with E( i ) = ad Var( i ) =, i = 1,...,, we have: RV EPECTED VALUE VARIANCE TCHEBSHEV'S BOUND Sum (S ) E(S )= Var(S )= 2 Mea ( ) E ( )=μ Var ( )= σ2 P ( S μ ε ) σ2 ε 2 P ( μ ε) σ 2 ε 2

21 STATISTICAL METHODS FOR BUSINESS UNIT 5. Joit aalysis ad limit theorems The Cetral Limit Theorem ad its applicatios

22 Cetral Limit Theorem (CLT) Levy-Lideberg's CLT : Give 1, 2,...,, iid radom variables with fiite E( i ) = µ ad Var( i ) = σ 2, i = 1, 2,,, the followig holds as : ( ) S = ¾¾ N μ,σ å i=1 i S-μ σ ¾¾ N(0,1) This approximatio is accurate for 30

23 Cetral Limit Theorem TOTAL EFFECT 1 2 S i1 i N,... MEAN EFFECT i1 i N, INDIVIDUAL CAUSES

24 Cetral Limit Theorem De Moivre's Theorem: Give 1, 2,...,, idepedet Beroulli RVs with parameter p, the followig holds as : i= 1 i p pq N (0,1) Cosequece: i i=1 Be ( p ) i B(,p ) N (p, pq ) A cotiuity correctio improves the accuracy of this approximatio

25 Approximatio BINOMIAL NORMAL = 5 = 20 = 40

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