multivariate random variable與joint distribution
我們可以個別調整每一種數組的出現程度高低。數學家命名為「多變量隨機變數」。理想的名稱應是「浮動數組」。
We can adjust the size of each array separately. Mathematicians are named " Multiple Variable Random Variables ". The ideal name should be " Floating Group ".
至於「每一種數組的出現程度高低」的函數，數學家命名為「聯合分布」。之前介紹了許多經典的分布，通通可以推廣成聯合分布。
As for the function "The extent to which each of these groups appears," the mathematician is named "Consolidated Distribution".
l={Line[{{3,3,1},{3,3,0}}], Line[{{5,6,1},{5,6,0}}], Line[{{8,2,1},{8,2,0}}], Line[{{2,8,1},{2,8,0}}]}; g=Show[ListPointPlot3D[{0,0,0}, Boxed -> False, Axes -> False, PlotRange -> {{0,10},{0,10}}], Graphics3D[{Thickness[0.01], CapForm["Butt"], Red, l}]] img=ImageResize[Rasterize[g, "Image", ImageResolution -> 72*3], Scaled[1/3]]; ImageCrop[RemoveBackground[img, {{{0, 0}}, 0.1}]]
F[x_, y_] :=PDF[MultinormalDistribution[{3, 3}, {{1, 0}, {0, 1}}], {x, y}] + PDF[MultinormalDistribution[{5, 6}, {{1, 0}, {0, 1}}], {x, y}] + PDF[MultinormalDistribution[{8, 2}, {{1, 0}, {0, 1}}], {x, y}] + PDF[MultinormalDistribution[{2, 8}, {{1, 0}, {0, 1}}], {x, y}]; Show[ Plot3D[F[x,y], {x, 0, 10}, {y, 0, 10}, PlotRange -> All, Boxed -> False, Mesh -> None, Axes -> False, PlotPoints -> 50, ColorFunction -> (ColorData["CherryTones"][Rescale[#3, {-4, 4}]] &)], ParametricPlot3D[{x, 2, F[x,2]}, {x, 0, 10}, PlotStyle -> Red] ] Plot[F[x,2], {x, 0, 10}, PlotRange -> All, PlotStyle -> Red, Axes -> False] Show[ Plot3D[F[x,y], {x, 0, 10}, {y, 0, 10}, PlotRange -> All, Boxed -> False, Mesh -> None, Axes -> False, PlotPoints -> 50, ColorFunction -> (ColorData["CherryTones"][Rescale[#3, {-4, 4}]] &)], ParametricPlot3D[Table[{x, y, F[x,y]}, {y, 0.2, 9.8, 0.2}], {x, 0, 10}, PlotStyle -> Red] ] Plot[Integrate[F[x,y],{y,-20,+20}], {x, 0, 10}, PlotRange -> All, PlotStyle -> Red]
數學家發明了一些特殊觀點，可以幫忙分析聯合分布：
Mathematicians have come up with a number of special points that can help analyse the distribution of the union:
條件分布 conditional distribution：部分維度只看特定數值。 例如(X=x,Y)是垂直截面，(X,Y=y)是水平截面。（乘上適當倍率，調整成分布。） 邊緣分布 marginal distribution ：只看部分維度。 例如X是疊加水平截面，Y是疊加垂直截面。
independent / uncorrelated
一個數組(x,y)，拆成兩個數字x與y，只有唯一一種方式。
One array (x,y) divided into two numbers x and y, the only way.
兩個數字x與y，併成一個數組(x,y)，只有唯一一種方式。
Two numbers x and y, merged into one array (x,y), the only way.
一個多變量隨機變數(X,Y)，拆成兩個隨機變數X與Y，只有唯一一種方式：邊緣分布。
A multivariant random variable (X, Y) which is split into two random variables X and Y, with only one method: edge distribution.
兩個隨機變數X與Y，併成一個多變量隨機變數(X,Y)，卻有無限多種方式：(X,Y)的分布擁有無限多種可能性。
Two random variables X and Y, combined into a multivariant random variable (X, Y), have unlimited distributions (X,Y) with unlimited multiplicity of possibilities.
Plot3D[Piecewise[{{0.25, 1 < x < 3 && 1 < y < 3}}], {x, 0, 4}, {y, 0, 4}, PlotRange -> {0,1}, ExclusionsStyle -> Directive[Thick, Red], Boxed -> False, Mesh -> None, Axes -> False, PlotPoints -> 50, ColorFunction -> (ColorData["CherryTones"][Rescale[#3, {-4, 4}]] &)] F[x_,y_] :=Piecewise[{ {0.15, 1 < x < 2 && 1 < y < 2}, {0.35, 2 < x < 3 && 1 < y < 2}, {0.35, 1 < x < 2 && 2 < y < 3}, {0.15, 2 < x < 3 && 2 < y < 3}}]; Plot3D[F[x,y], {x, 0, 4}, {y, 0, 4}, PlotRange -> {0,1}, ExclusionsStyle -> Directive[Thick, Red], Boxed -> False, Mesh -> None, Axes -> False, PlotPoints -> 50, ColorFunction -> (ColorData["CherryTones"][Rescale[#3, {-4, 4}]] &)]
數學家從中挑選比較特別的款式，方便討論與運用：
Mathematicians choose a more special style from which to discuss and use:
獨立 independent ：p(X,Y)(x,y)=pX(x)pY(y) 不相關 uncorrelated：E[XY]=0
數學家的敘述方式是「兩個隨機變數X與Y是獨立的」，背後意義是「多變量隨機變數(X,Y)的分布是一種特殊款式：獨立」。
Mathematicians describe it as "two random variables X and Y are independent", followed by "the distribution of multivariant random variables (X, Y) is a special formula: independence".
p={{1, 1, 0.05}, {1.5, 1, 0.1}, {-1, -1, 0.1}, {-1, 0, 0.05}, {0, 1.5, 0.05}, {0, 0, 0.1}, {-1, -1.5, 0.2}, {1, -2, 0.15}, {-2, 1, 0.2}}; l=Table[Line[{p[[i]], {p[[i,1]], p[[i,2]], 0}}], {i,1,9}]; g=Show[ListPointPlot3D[{0,0,0}, Boxed -> False, Axes -> False, DataRange -> {{-2,2},{-2,2}}, PlotRange->{0,0.4}], Graphics3D[{Thickness[0.01], CapForm["Butt"], RGBColor[192,0,0], l}]] img=ImageResize[Rasterize[g, "Image", ImageResolution -> 72*3], Scaled[1/3]]; ImageCrop[RemoveBackground[img, {{{0, 0}}, 0.1}]]
px={{2, 0.6}, {3, 0.25}, {2.5, 0.15}}; py={{3, 0.6}, {2, 0.25}, {1 , 0.15}}; pxy={{2, 3, 0.6*0.6}, {2, 2, 0.6*0.25}, {2, 1, 0.6*0.15}, {3, 3, 0.25*0.6}, {3, 2, 0.25*0.25}, {3, 1, 0.25*0.15}, {2.5, 3, 0.15*0.6}, {2.5, 2, 0.15*0.25}, {2.5, 1, 0.15*0.15}}; l=Table[Line[{pxy[[i]], {pxy[[i,1]], pxy[[i,2]], 0}}], {i,1,9}]; g=Show[ListPointPlot3D[{0,0,0}, Boxed -> False, Axes -> False, PlotRange -> {{0,4},{0,4}}], Graphics3D[{Thickness[0.01], CapForm["Butt"], RGBColor[192,0,0], l}]] img=ImageResize[Rasterize[g, "Image", ImageResolution -> 72*3], Scaled[1/3]]; ImageCrop[RemoveBackground[img, {{{0, 0}}, 0.1}]]
F[x_] :=PDF[GammaDistribution[2, 1], x]; G[y_] :=PDF[NormalDistribution[2, 0.5], y]; Plot[F[x], {x, 0, 4}, PlotRange -> {0,1}, Filling -> Axis] Plot[G[y], {y, 0, 4}, PlotRange -> {0,1}, Filling -> Axis] Show[ Plot3D[F[x]*G[y], {x, 0, 4}, {y, 0, 4}, PlotRange -> All, Boxed -> False, Mesh -> None, Axes -> False, PlotPoints -> 50, ColorFunction -> (ColorData["CherryTones"][Rescale[#3, {-4, 4}]] &)], ParametricPlot3D[Table[{x, y, F[x]*G[y]}, {y, 0.1, 3.9, 0.1}], {x, 0, 4}, PlotStyle -> Red] ]
獨立：(X=x,Y=y)的出現程度，等於X=x的出現程度、Y=y的出現程度兩者相乘。
Independence: (X=x, Y=y) equals X=x and Y=y multiplied.
所有垂直截面皆相似、所有水平截面皆相似。
All vertical cross-sections are similar and all horizontal cross-sections are similar.
條件分布、邊緣分布，兩者相似（僅倍率不同，可以是零倍）。
Conditional distributions, edge distributions, are similar (only multiples, they can be zeros).
無論有X沒X，都不影響Y的分布形狀，因而稱作獨立。
X without X does not affect the distributional shape of Y and is therefore called independence.
數學家很喜歡假設隨機變數是獨立的，讓分布變得很漂亮。
Mathematicians like to assume that random variables are independent, making distribution beautiful.
F[x_] :=PDF[StudentTDistribution[1], x]; G[y_] :=PDF[StudentTDistribution[1], y]; a=Cos[Pi/4]; b=Sin[Pi/4]; Plot3D[F[a*x-b*y]*G[b*x+a*y], {x, -5, 5}, {y, -5, 5}, PlotRange -> All, Boxed -> False, Mesh -> None, Axes -> False, PlotPoints -> 50, ColorFunction -> (ColorData["CherryTones"][Rescale[#3, {-4, 4}]] &)]
不相關：共相關數等於零。X與Y的點積是0。
Not relevant: the sum is equal to zero. The X and Y dots are zero.
共相關數 E[XY] ：每個數組的XY相乘，求平均數。 正相關 E[XY] > 0：一三象限比較厚實。大致成正比。 負相關 E[XY] < 0：二四象限比較厚實。大致成反比。 不相關 E[XY]=0：一三象限、二四象限，勢均力敵。
獨立較嚴格，不相關較寬鬆。獨立僅一種，不相關有多種。
Independence is more stringent and less convoluted. Independence is only one, not many.
有件事值得一提：當X或Y的平均數是零，獨立導致不相關。白話解釋是左右等量或者上下等量。
One thing is worth mentioning: when the average number of X or Y is zero, independence is irrelevant. The white interpretation is either left or right or upper or lower.
independent=> uncorrelated if E[X]=0 or E[Y]=0
E[X]=sum x p(x)
?
E[Y]=sum y p(y)
?
E[XY]=sum sum x y p(x,y)
? ?=sum sum x y p(x) p(y) 獨立
? ?=sum sum x p(x) y p(y)
? ?=(sum x p(x)) (sum y p(y)) 交叉相乘
? ?=E[X]E[Y]=0
不相關的定義有兩種：共相關數是零、共相關係數是零。後者直接得到「獨立導致不相關」的結論，省去「當平均數是零」的限制。
There are two non-relevant definitions: the number is zero, the number is zero, and the number is zero. After that, one gets the conclusion that "independence is irrelevant" and saves the limit of "when the average is zero".
definition of "uncorrelated"
E[XY]=0
another definition of "uncorrelated"
E[(X-E[X])(Y-E[Y])]
————————————————————————————=0
√ E[(X-E[X])2] E[(Y-E[Y])2]
transformation of multivariate random variable
Plot3D[F[x/1.2,y/0.6], {x, 0, 10}, {y, 0, 10}, PlotRange -> All, Boxed -> False, Mesh -> None, Axes -> False, PlotPoints -> 50, ColorFunction -> (ColorData["CherryTones"][Rescale[#3, {-4, 4}]] &)] t=Pi/6; Plot3D[F[x*Cos[t]-y*Sin[t], x*Sin[t]+y*Cos[t]], {x, 0, 10}, {y, 0, 10}, PlotRange -> All, Boxed -> False, Mesh -> None, Axes -> False, PlotPoints -> 50, ColorFunction -> (ColorData["CherryTones"][Rescale[#3, {-4, 4}]] &)]
多變量隨機變數可以實施函數變換。雙射函數，浮動數字一齊改變，出現程度互不干涉，機率不必增減。
Multivariate random variables can be used to effect a function change. Double-launched functions change the floating numbers so that they do not interfere with each other. The probability does not need to be increased.
translate: (X+a, Y+b)
scale: (sX, tY)
rotate: (Xcos?θ-Ysin?θ, Xsin?θ+Ycos?θ)
多變量隨機變數可以實施仿射變換，變成不相關；但是無法變得獨立，只能變得盡量獨立。請見本站文件「principal component analysis」。
Multivariant random variables can be imitated and become irrelevant; but they cannot become independent and can only be as independent as possible. See the site document "principal component analysis.
獨立、不相關的多變量隨機變數，實施仿射變換，性質可能保留或失效。
Independent, unrelated multivariant random variables, i.e. imitation variants, which may be retained or invalidated.
獨立 p(X,Y)(x,y)=pX(x)pY(y) 位移：仍然獨立 p(X,Y)(x+a,y+b)=pX(x+a)pY(y+b) 縮放：仍然獨立 p(X,Y)(sx,ty)=pX(sx)pY(ty) 旋轉：通常失效 p(X,Y)(xcos?θ-ysin?θ,xsin?θ+ycos?θ) 不相關 E[XY]=0 位移：通常失效 E[(X+a)(Y+b)]=E[XY] + aE[Y] + bE[X] + ab 縮放：仍然不相關 E[(sX)(tY)]=stE[XY]=0 旋轉：通常失效 E[(Xcos?θ-Ysin?θ)(Xsin?θ+Ycos?θ)]=(E[X2]-E[Y2])sin?θcos?θ
random vector
多個隨機變數，合稱「隨機向量」。
Multiple random variables, commonly known as " random vectors ".
一個隨機變數推廣成多個隨機變數，衍生歧義：
A random variable spreads to multiple random variables, which are derived from the following:
多變量 multivariate ：一個變數推廣成一個數組。重視聯合分布。 多變數 multivariable：一個變數推廣成多個變數。輕視聯合分布。
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