3.1二维随机变量及其分布

3.1二维随机变量及其分布 3.1二维随机变量及其分布 In many practical problems In many practical problems, in many practical problems In many practical problems, the results of randomised trials are often needed The results of randomised trials often require the results of randomised trials The results of randomised trials are often needed It's described by multiple random variables Use multiple random variables to describe multiple random variables Use a number of random variables to describe, such as the position of the target at the target For example, the position of the target when hitting the target, such as the position of the target For example, the position of the target at the target You have to use the x-coordinate You need to use the x-coordinate You have to use X and y coordinates And the y-coordinate and the y-coordinate And Y to determine that To be sure To be sure, they're defined in the same way They are defined in the same way as they are defined They're defined in the same way A sample space A sample space is a sample space A sample space, S = {e} = {all the points All the bullet points Two of all the bullet points Two of the top two On the two A random variable Random variable random variable Random variables. Again Like and like Again, sinusoidal ac voltage requires the use of amplitudes The sine ac voltage needs to be used with the amplitude sine ac voltage The sine ac voltage needs to be used in amplitude, ,,, , frequency and Frequency and frequency The frequency and In the early phase In the early phase in the early phase The initial phase three real Numbers are described A real number that describes a real number This is a real number This is the This is 3 random variables A random variable, a random variable A random variable. These random variables These random random random things These random first The first the first Chapter 3 chapter Chapter multidimensional random variables and their distributions Multidimensional random variables and their distributed multidimensional random variables and their distributions The multidimensional random variable and its distribution at the beginning In the early phase in the early phase The initial phase three real Numbers are described A real number that describes a real number This is a real number This is the This is 3 random variables A random variable, a random variable A random variable. These random variables These random random random things These random Between the variables Between variables There is a general connection between variables In general, there is some kind of connection that is usually associated with some kind of connection Generally speaking, there is a connection, so you need to get them So you need to take them and you need to have them So you need to have them Think of it as a whole Think of it as a holistic approach to research as a whole Think of it as a whole, and that's what we're going to talk about So that's what we're going to talk about and that's what we're going to do That's what we're going to talk about Multidimensional random variables Multidimensional random variables of multidimensional random variables Multidimensional random variables. For simplicity For brevity, for brevity For the sake of simplicity, we're only talking about two-dimensional random We're just talking about two dimensional random and we're talking about two dimensional random We're just talking about two-dimensional random Variable situation The condition of the variable In the case of a variable, the n dimensional random variable can do the same thing The dimension random variable can do something like this and we can do a similar discussion about the dimension random variable The dimension random variables can be discussed in a similar way. 3.1 two-dimensional random variables and their distributions Two-dimensional random variables and their distribution of two-dimensional random variables and their distributions Two-dimensional random variables and their distributions define By defining Definition 3.1 3.13.1 3.1 is a random variable Let's set random variables to be random variables Let's say that the random variable X is equal to X of e And and And Y = Y (e) It's two random variables that are defined in the same sample space The two random variables that are defined in the same sample space are two random variables that are defined in the same sample space It's two random variables that are defined in the same sample space, ,, , It's a vector that they make up A vector that is made up of them is a vector that is made up of them It's a vector (X (e), Y (e)) that they make up Let's call it two-dimensional Called two-dimensional 3.1.1. 3.1.1.3.1.1. 3.1.1. The definition of a two-dimensional random variable and its distribution function The definition of a two-dimensional random variable and the definition of a two-dimensional random variable and its distribution function The definition of a two-dimensional random variable and its distribution function It's a vector that they make up A vector that is made up of them is a vector that is made up of them It's a vector (X (e), Y (e)) that they make up Let's call it two-dimensional Called two-dimensional Random vector Random vector random vector Random vectors or two-dimensional random variables Or two dimensional random variables or two dimensional random variables Or two dimensional random variables. .. It is. Accordingly, said Correspondingly, it is said So let's call this the formula for X, Y The law of the values of the values of the values The rules of the values are two-dimensional and random Two dimensional random variable two-dimensional random variation Two dimensional random variable The amount measure The distribution of quantity (X, Y) Distribution of distributions The distribution of ,, It's the same as the one-dimensional case It's the same as the one-dimensional case In the same way as one dimension, ,, We can also We can, too We can also To do this with distributed functions The distribution function is used to study the distribution function To study the values of two-dimensional random variables in terms of distribution functions The two-dimensional random variable evaluated by a two-dimensional random variable Two dimensional random variables regular Rule of law The rule. .. It is. define By defining Definition 3.2 3.23.2 3.2: : : Set: Set a Let's say (X, Y) is a two-dimensional random variable The two-dimensional random variable is a two-dimensional random variable It's a two-dimensional random variable, ,, For any Arbitrary to any one For any (x, y) ? ? ? ? R2. said Is said is said It's called the binary function Function of two variables Dual function F (x, y) = P {(x is less than or equal to x) Or less or less Less than or equal to x Or less or less It's less than or equal to P {X is less than or equal to X Or less or less Or less x, Y or less Or less or less Y} or less for As for the For X, Y Of the The distribution function Distribution function Distribution function, ,, Or, Or or Or X and And with the With Y Of the The joint distribution function Joint distribution function Joint distribution function. .. It is. Pay attention to Pay attention to the notice Note: : : : The two-dimensional distribution function is defined on the whole plane The two-dimensional distribution function is defined on the entire plane by the two-dimensional distribution function The two-dimensional distribution function is defined on the whole plane, ,, What, By the the In order to find the two-dimensional distribution function In order to find a two-dimensional distribution function in order to find a two-dimensional distribution function In order to find the two-dimensional distribution function, ,, In the whole plane You have to do it on the full plane It's going to be on the full plane Feel secure Worry about Lv. .. It is. Geometric interpretation Geometric interpretation of geometry Geometric interpretation: : : If you view a two-dimensional random variable as a plane If you view a two-dimensional random variable as a plane if you view a two-dimensional random variable as a plane If you view a two-dimensional random variable as a plane Machine of the coordinates of the point The coordinates of the coordinates of the machine point The coordinates of the machine, ,, , then So then So, ,, The distribution function Distribution function The distribution function F of x, y is there In the It's in (x, y) In place of Place the The function is just random points The value of the function is just random points The function value is the random point (x, Y) falls below the point shown below The points that fall below are shown in the image below The point (x, y) shown below. The probability of the vertex at the lower left of the point The probability of the vertex in the region at the lower left of the point is the probability of the region at the lower left of the point The probability of the vertex at the lower left of the point. .. It is. (x2, y2) (x1, y2) 1212 () {,} XYxxxyyy < is less than or less than or equal to the figure above The probability for 1212 22211211 {and} () () () PxXxyYy FxyFxyFxyFxy < < or less or less =?? + (x1, y1) (x2, y2) (x2, y1) (x1, y2) distribution function Distribution function The distribution function F of x, y is the following The following is as follows Have the following properties Nature of nature The properties: 0), (lim), (= =? Infinity? Infinity, infinity, yxFFx, (), 1x YFFxy goes to plus infinity plus infinity plus infinity is equal to (1) the normative Normative normative The normative is arbitrary For any one For any (x, y) ? ? That's the same thing as R2, 0 Or less Or less or less Less than or equal to F of x, y is less than or equal to Or less or less 1, or less? The up - ? The up - y X (), () 0. Up? Up = = For any fixed (,), lim (). Up? Infinity = = is equal to any fixed. (2) F (x, y) is a variable Variables are variables Is a variable x or Or or Or the monotonic function of y The monotonic nondecreasing function of monotonic function The monotonic nondecreasing function For any For any one For any y ? ? ? ? R, when dangdang When the x1 < x2 All the time When, ,, , F (x1, y) Or less Or less or less It's less than or equal to F of x2, y. ;; ; For any For any one For any x ? ? ? ? R, when dangdang When y1 < y2 All the time When, ,, , F (x, y1) Or less Or less or less Less than or equal to F of x, y2. Y, x (F), y, x (Flim) y, 0x (F0) xx 00 = = = = = = = = + + + + + + + + - - - Y, x (F) y, x (Flim) yy 00 = = = = = = = = + + + + + + + + - - - (3) F (x, y) About about With respect to x or Or or Y right or continuous Continuous right continuous right The right is continuous to any For any one For any x ? ? ? So, the "R", "y" ? ? In the case of the, R, (4) rectangle inequality Rectangle inequality Rectangular inequality For any For any of these For any one (x1, y1), (x2, y2) ? ? ? The sum of R2, x1, x2, ,, , y1 < y2), F (x2, y2) - -- Minus F of x1, y2, -- Minus F of x2, y1, plus + + Plus F of x1, y1 is greater than or equal to P p Acuity 0. Conversely Vice and vice On the other hand, ,, Any of the two functions that satisfy these four properties Any binary function that satisfies the four properties of each of these four properties is satisfied Any two function that satisfies the four properties of the above is the reverse Vice and vice On the other hand, ,, Any of the two functions that satisfy these four properties Any binary function that satisfies the four properties of each of these four properties is satisfied Any two function that satisfies these four properties F (x, y) can be a two-dimensional random variable Both of these can be used as a two-dimensional random variable It could be a two-dimensional random variable (X, Y). The distribution function Distribution function of the distribution function Distribution function. .. It is. case Cases of cases Case 1. 1.1. We know the two-dimensional random variable We know that the two-dimensional random variable is known as a two-dimensional random variable We know the distribution function of the two-dimensional random variable (X, Y) The distribution function is the distribution function The distribution function is)] 3 ()] [ 2 ([), ( y arctgC x Arctgbayxf + + = 1) the constant A, B, C. P {0 < 2, 0 < Y < 3} () [] 1 22 FABC PI + infinity + infinity = + = 0 y ArctgCBAyF PI 0)] 3 (] [ 2 (= +? =? Infinity y ArctgCBAyF PI 0] 2 ] [ 2 ([) = PI C x arctgBAxF21 2 PI. PI = = =? ACB1 {02, 03} (2, 3) (0, 3) (2, 0) (0, 0) 16 PXYFFFF < less than or equal to =? + = definition By defining Definition 3.3: : : If you have a two-dimensional random variable If the two-dimensional random variable is a two-dimensional random variable If the two-dimensional random variable (X, Y) (X, Y) (X, Y) (X, Y) can only be taken up to the maximum You can only take as much as you can You can only take as much as you can Column value Make a list of values I'm going to write the value of xi, yj = = = = 1, 2... ), ,, Is said, Is said is said Let's call it X, Y. (X, Y) (X, Y) (X, Y) as the As for the For the second . The second Discrete random variables The discrete random variable of the dimension discrete variable Discrete random variables. .. It is. 3.1.2 3.1.2 3.1.2 The probability distribution of the 3.1.2 two-dimensional discrete random variable The probability distribution of the two-dimensional discrete random variable is the probability distribution of the two-dimensional discrete random variable The probability distribution of a two-dimensional discrete random variable is a two-dimensional discrete random variable If the two-dimensional discrete random variable is a two-dimensional discrete random variable If you take the two-dimensional discrete random variable (X, Y) Take take The probability of xi, yj The probability of probability is zero The probability of p pp pij ijij Ij, ,, Is said, Is said is said Says P {X = = = = xi, Y = = = = yj,} = = = = pij, ,, , ij ijij Student: ijP = = = 1, 2... ), ,, , It's a two-dimensional discrete random variable The two-dimensional discrete random variable is a two-dimensional discrete random variable For the two-dimensional discrete random variable (X, Y) Of the The distribution law of Distribution law Distribution law, ,, , or random Or random or random Or random variable Variable variables Variables X and And with the With Y Of the The joint distribution law The joint distribution law The joint distribution law. .. . But remember to But remember that it is But remember to (X, Y) - ~ ~ ~ P {X = = = = xi, Y = = = = yj,} = = = = pij, ,, , (I, j = = = = 1, 2... X Yy one y 2 ... y J... P11 p12... P1j... P21 p22... P2j... ... ... ... ... The distributive law of the x1, x2, two-dimensional discrete random variable is also listed as follows The distributive law of two-dimensional discrete random variables can also be listed as follows for the distribution of two-dimensional discrete random variables The distributive law of two-dimensional discrete random variables is also listed as follows: Pi1 pi2... Pij... ... ... ... ... ... ... ... ... The properties of the joint distribution law The properties of the joint distribution law The properties of the joint distribution law (1) pij is greater than 0, I, j = 1, 2,... ; (2) 1 p1i1j Ij = ? ? ? ? ? ? ? ? acuity P p P p P p Xi cases or higher Cases of cases Case 2. 2. 2. There are two red balls in the bag There are two red balls in the bag There are two red balls in the bag, ,, , three white balls Three white balls and three white balls Three white balls, ,, I don't want to touch the ball now No return to touch the ball now No return to touch the ball secondary Two times two times Second, ,, , To make Make? ? ? = ? ? ? = Touch the white ball for the second time Touch the red ball for the second time First touch the white ball First touch the red ball 0 one 0 one Y X, The distribution of X, Y. 2 5 2 2} 1, 1 { P P YXP = = = 32 x? ? Touch white ball 0X for the second time Y one 0 1, 010 one 10 3 10 3 10 32 53 2 } 0, 1 { P YXP x = = = 2 52 3 1, 0} { P YXP x = = = 2 5 2 3} 0, 0 { P P The joint distribution density of the 3.1.3 two-dimensional continuous random variable The joint distribution density of two dimensional continuous random variables in a 2d continuous random variable The joint distribution density definition of a two-dimensional continuous random variable By defining Define 3.4 for the two-dimensional random variable For two dimensional random variables for two dimensional random variables For the two-dimensional random variable (X, Y), ,, , if any If existence exists If there is A non-negative integrable function A non-negative integrable function, a non-negative integrable function A non-negative integrable function f of x, y, ,, To make the To make the Made of? ?? ? (x, y) ? ? ? ? R2. ,, , The distribution function Its distribution function is its distribution function Its distribution function is the integral ? ? ? ? ? ? ? up Up up Up? ?? ? up Up up Up? ?? ? = = = = x Y, Du dv, v, u of f, y, x of f infinity Up up Up? ?? ? up Up up Up? ?? ? said Is said is said It's called (X, Y) as a two-dimensional continuous random variable The two-dimensional continuous random variable is a two-dimensional continuous random variable For a two-dimensional continuous random variable, ,, , f (x, y) as As for the for The density function of X, Y The density function of the density function Density function (probability density Probability density The probability density, ,, Or, Or or Or X and And with the With Y Of the The joint secret Joint secret Joint secret Degree of function The degree function Degrees of function, ,, , but remember to But remember that it is But remember to (X, Y) ~ ~ F of x, y, ,, (x, y) ? ? So, it's the same thing as R2 (1) Non-negative nonnegative Nonnegative: f (x, y) is greater than or equal to P p It's greater than 0, (x, y) ? ? ? R2; (2) normative Normative normative Standardization: And vice Vice and vice On the other hand, ,, There are two functions of two functions The binary function with the above two properties has two functions The function f of x, y, with these two properties, ,, That will be Will be Will (1), Fxydxdy, up and up ? Up? Infinity is the integral of the integral Joint density combined density The properties of the joint density f (x, y) The nature of the property The nature of); Y, x (f yx ) y, x (F2 = = = = ? ?? ?? ?? ? ? ?? ? And vice Vice and vice On the other hand, ,, There are two functions of two functions The binary function with the above two properties has two functions The function f of x, y, with these two properties, ,, That will be Will be will It's the density function of a two-dimensional continuous random variable The density function of a two-dimensional continuous random variable is the density function of a two-dimensional continuous random variable It's the density function of a two-dimensional continuous random variable. .. It is. (3) if If if If f (x, Y) in In the In the (x, y) ? ? ? R2 in continuous It's contiguous Continuous, ,, , there are Have had There are In addition In addition, moreover In addition, ,, F of x, y, and the following properties And the following properties also have the following properties There are also the following properties of the integral of the integral from the integral of the integral to the region of the plane For any plane region, for any plane region For any plane region G? ?? ? R2, case 3? < < < < = Yx10, 101 ? ?? < < < < = others yx yxfYX 0 10101 (~) : P {X > Y} 1 001 {} 1 2 xpxydxdy > =? = ? ? o please O: : : (1) (1) (1) (1) constant Constant constant Constant A AA A; ;; ; (2) (2) and (2) (2) F (1, 1); ;; ; (3) (X, Y) (3) (X, Y) (3) (X, Y) (3) (X, Y) falls in the triangle area Fall in the triangle area and fall in the triangle area It falls in the triangle region D DD D: : : P: x P p Acuity 0, y p P p Acuity 0, 2 x + 3 y or less Or less or less 6 or less The probability of the The probability of the inside probability In probability. .. It is. ? ? ? > > = + ? Other, 0 0, 0, ), (~), () 32 (yxAe yxfYXy X example Cases of cases Set example 4. Set a set solution solving The solution (1) is normative The normative is the norm By normative 6 =? A11 (23) 23 00 (2) (1, 1) 6 (1) (1) (1) xyFedxdyee? +?? = =?? ? ? (23) 00 () ? + Infinity is equal to the integral from the integral of (3) (X, Y) (3) (X, Y) (3) (X, Y) falls in the triangle area Fall in the triangle area and fall in the triangle area It falls in the triangle region D DD D: : : X: xx X p P p Acuity 0, y Zero, y0, y 0, y p P p Acuity 0, 2 x + 3 y 0, 2X plus 3y0, 2X plus 3y 0, 2 x + 3 y or less Or less or less 6 or less Six, six 6 The probability of the The probability of the inside probability In probability. .. It is. Solution of dx dy eDYXPD Yx ? ? + ? () 2. +? = 3 3 2 2 32 (6) dyedxx Yx ? ? +? = 00 32 (6) Dyedxyx671?? = e () () , 0, 0 (,) 0, (1); (2) ().xyaexy fxy AFxy G factor eta Factor eta? +? > > = ? ? Example 5 the two-dimensional random variable has the probability density other Let's find the distribution function Take the probability of falling in the figure below (3).g xi eta The probability of falling in the figure below is 34Gxy: solution, 1), () 1 (= the integral from the integral of the integral from 0 to infinity Up? + up Up? Dx dy Yxf by the nature of the probability density 1 = ? AAeeAdxdyAeyxyx =?? = = up +? Up +? + + up up +? 00 ? ? 00 (a) [] [] [] ? ? ? ? > > = = ? ? ? ? + ? Up? Up? other . 0 0, 0, (), (), () () () () () () () () () () () () () (y) xdudve dudvvufyxFxy vu xy1 = ? A other 0, 0 , , 0 ) (1, 1) > > ? ? ? ?? =? ? yx eey X (3) (()) 43 (a) 4 000.8558 x Xydxedy?? + = = the integral from the integral of 34Gx1 2d evenly distributed two-dimensional uniform distribution Two dimensional uniform distribution G is a bounded area on the plane It's the bounded area on the plane that is the bounded area on the plane It's a bounded area on the plane Its area is its area The area is A, if it's in two dimensions If two dimensions are followed by two dimensions If the 2 d with Machine variable Variable of machine variable The density function of the machine variable (X, Y) The density function of the density function is zero The density function is 21 (,) (,) xyGR fxy a. , i. ? ? Weng? = i. ? ? Two commonly used two-dimensional continuous distributions Two commonly used two-dimensional continuous distributions of two commonly used two-dimensional continuous distributions Two commonly used two-dimensional continuous distributions 0, others i. ? ? ? ? {(,)} D GS PXYD s. ? = easy to see Easy to see easy to see Easy to see, ,, If, If if If ( (( (X, ,, , Y) )) ) in the area It's in the region In the area of G Up and (on the inside nene The inside) follows the uniform distribution The uniform distribution follows uniform distribution Uniform distribution , ,, for yeah For any region of G Any region in any region Any region of the region D, ,, , there are There are There are said Is said is said I'll call it (X, Y) in the region It's in the region In the area of G Up and (on the inside nene The inside) follows the uniform distribution The uniform distribution follows uniform distribution Obey the uniform distribution. .. It is. (2) the two-dimensional normal distribution Two dimensional normal distribution of normal distribution Two dimensional normal distribution If you have a two-dimensional random variable If the two-dimensional random variable is a two-dimensional random variable The density function of the two-dimensional random variable (X, Y) The density function of the density function is zero The density function is 2 one one 2 2 21) [( 1) (2 one Exp { 12 one A), (sigma mu rho Rho sigma PI sigma? ? ? ? = x Yxf]} () 221 sigma mu sigma mu sigma mu Rho? + ?? ? yyx22 1212 (,) N Mu mu mu sigma rho ()) (2221 sigma sigma Rho + ? ) "(~), (2) 2 2 121 rho Sigma mu of mu NYX One of which Among them, ,, , mu Mu mu Mu 1. ,,, , mu Mu mu Mu 2 for real For real Numbers For real, sigma Sigma sigma Sigma 1 > 0, ,,, And the sigma Sigma sigma Sigma 2 > 0, ,,, , | rho Rho rho Rho | < 1, ,, , Then, the According to weigh The (X, Y) is subject to the parameter The obedience parameter is subject to the parameter The obedience parameter is mu Mu mu Mu 1, mu Mu mu 2, mu sigma Sigma sigma Sigma 1 2, sigma Sigma sigma Sigma 2 2, rho Rho rho The two dimensional normal of rho The two dimensional normal state of the two-dimensional normal The two-dimensional gaussian distribution The distribution of distribution The distribution, ,, , but remember to But remember that it is But remember to 12 121122,,, () {,} n NNNXXXN FxxxPXxXxXx = or less or less or less L LL For any n real variables, we call that Known as known as It's called the n dimensional random variable Dimension random variable The distribution function of the dimension random variable (X1, X2,... Xn) The distribution function or the distribution function that follows The distribution function or whatever Machine variable Variable of machine variable X1, X2, The joint distribution function of Xn The joint distribution function of the joint distribution function The joint distribution function, ,, It has the class It has classes it has classes It has the class The property of a distribution function that looks like a two-dimensional random variable The properties of the distributed functions of a two-dimensional random variable are similar to the properties of the distributed functions of the two-dimensional random variables The property of a distribution function that looks like a two-dimensional random variable.