3.2MarginaldistributionMarginal distribution function for bivariateDefine-P57lm,F(cP[X≤x)= F (X, +o) =lim F(x,y)= P[Y≤y)Fy(y) = F (+oo, y) =X-→+00the marginal cdfs of (X, Y with respect to X andY respectively
FY(y)＝F (+, y)＝ ＝P{Yy} lim 𝑦→+∞ 𝐹(𝑥, 𝑦) lim 𝑥→+∞ 𝐹(𝑥, 𝑦) 3.2 Marginal distribution (x)＝F (x, +)＝ ＝P{Xx} Marginal distribution function for bivariate Define –P57 the marginal cdfs of (X, Y) with respect to X and Y respectively
Example 1. Suppose that the joint distributionof (X,Y) is specified by1-e-x-xe-y0≤x≤yF(x,y) =1-e-v-ye-y0≤y≤x0elseDetermine Fx(x) and Fy(y)Ans wer:4x≥0Fx(x)=F(x,800x<0"-ye-yJ≥029Fy(y)-F(c0,y)-0J<0
Example 1. Suppose that the joint distribution of (X,Y) is specified by 1 0 ( , ) 1 0 0 x y y y e xe x y F x y e ye y x else − − − − − − = − − Determine FX(x) and FY(y)。 Answer: FX(x)=F(x,)= − − 0 0 1 0 x e x x FY(y)=F(,y)= − − − − 0 0 1 0 y e ye y y y
Marginaldistributionfor discretedistributionSuppose that(X, Y) ~ P(X = Xi, Y = yj,} = pii , i, j = 1, 2, ...Define-P57. i=1, 2, ...P(X = xi} = Pi. =Z1 pijP(Y = yif = p.j=. j=1, 2, ..i≥1Pijthe marginal pmf of (X, Y) with respect to X and Yrespectively
Marginal distribution for discrete distribution Suppose that (X, Y)～ P{X＝xi , Y＝ yj ,}＝ pij ，i, j＝1, 2, . Define-P57 P{X＝xi }＝pi .＝ ，i＝1, 2, . P{Y＝ yj }＝p.j＝ ，j＝1, 2, . 𝑗≥1 𝑝ij 𝑖≥1 𝑝ij the marginal pmf of (X, Y) with respect to X and Y respectively
MarginaldensityfunctionSuppose that (X, Y) ~f (x, y), (x, y)eR2, definefr(y) = / f(x,y)dxfx(x) = / f(x,y)dythe marginal pdf of (X,Y) with respect to X and Y.Example3.4-P59
Marginal density function the marginal pdf of (X,Y) with respect to X and Y. Example 3.4-P59 𝑓𝑋(𝑥) = න 𝑓(𝑥, 𝑦)dy 𝑓𝑌(𝑦) = න 𝑓(𝑥, 𝑦)dx Suppose that (X, Y)～f (x, y), (x, y)R 2 , define
Fy(y) = F(0, y) =f(x,y)dx dyfr(y) = / f(x,y)dxthe marginal pdf of Y
𝑓𝑌(𝑦) = න 𝑓(𝑥, 𝑦)𝑑𝑥 . the marginal pdf of Y 𝐹𝑌(𝑦) = 𝐹(∞, 𝑦) = න න 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦