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MindMap Gallery Chapter 3 Multidimensional random variables and their distribution

Chapter 3 Multidimensional random variables and their distribution

This is a mind map about multi-dimensional random variables and their distribution in Chapter 3. The main content includes: theme, key content, Section 4. Distribution of two random variable functions Z=g(X,Y), Section 3 Two-dimensional uniform distribution and two-dimensional normal distribution, Section 2: Independence of random variables (detachable), Section 1: Two-dimensional random variables and their distribution.

Edited at 2024-03-31 23:16:52

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Chapter 3 Multidimensional random variables and their distribution

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Outline

Chapter 3 Multidimensional random variables and their distribution

Important content

Two-dimensional random variable (X, Y) (understood as the product event about X and about Y (satisfied simultaneously))

Necessary and sufficient conditions for distribution law

Continuous random vector (X,Y)

Necessary and sufficient conditions for distribution function

Necessary and sufficient conditions for density function

The probability of a continuous random vector = the integral of the corresponding density function over the corresponding interval

difficulty

Edge density function calculation method (discussed in sections)

1. Find the support (yellow area) and the joint density function is not 0 -> get the corresponding x value range

2. Find the edge density function for x --->fix x, run it once for y from -∞ to ∞ (integration limit)

3. Process the endpoints (note the range of support f(x,y)>0)

Conditional density function calculation method

1. Find the edge density function of (X,Y) about X and about Y

2. Handling domains and endpoints

1. According to the condition X(Y)=? Determine the range of the condition X(Y) in the domain of the edge density function

2. Are you sure that the condition X(Y)=? Below, the definition range of another variable Y(X)

3. Joint density function/edge density function (corresponding conditions) The definition area is written separately (more clear)

Two-dimensional normal distribution

Rotation properties (look back later)

summary

Two-dimensional normal--->marginal distribution normal (marginal distribution normal--//--> two-dimensional normal)

X, Y are both normal and independent ---> (X, Y) normal

(X,Y)Normal X,Y are independent<--->X,Y are not related

(X,Y) normal <---> (aX bY, cX dY) normal and coefficient determinant ≠0

Distribution of functions of random variables

Section 1 Two-dimensional random variables and their distribution

two-dimensional random variable

introduction

X and Y cannot take values freely respectively (X and Y are not independent)

Only by studying the whole (X, Y) can we reflect the relationship between X and Y

definition

Testable mapping of S->R2

Distribution function of (X,Y)

Purpose (to study the distribution of (X, Y) as a whole)

definition

nature

(X,Y) distribution function necessary and sufficient conditions

1. F(x,y) is monotonic with respect to x and with respect to y

2. F(x,y) is right continuous with respect to x and with respect to y

3. Special value

Corresponding to the necessary and sufficient conditions for one-dimensional random variables

F{x1≤X≤x2,y1≤Y≤y2} (x, y with the same subscript are positive, and the cross term is negative)

(X,Y) marginal distribution about X and about Y

definition

meaning

Let y->positive infinity, then {y takes all values} is an inevitable event

Without restrictions on y, find the probability that X satisfies x

Discrete two-dimensional random vector

The joint distribution law of (X,Y)

P{X=xk,Y=yk}

significance

Find the probability of an event that satisfies the product of X and Y simultaneously

Seeking the law

Solving based on the event itself

multiplication formula

definition

The values of (X,Y) can be listed at most

express

sheet

Properties (necessary and sufficient conditions for distribution law)

non-negativity

normative

Joint distribution function of (X,Y)

F(x,y)

Marginal distribution law of (X,Y)

Definition (for fixed xi, summation formula for yj (no restriction on y))

Multiplication formula Total probability formula

conditional distribution law

definition

Find the probability of Y=yj under the condition that X is xi (P{X=xi}>0)

Properties (satisfying the necessary and sufficient conditions of distribution law)

non-negativity

Normativity (under the condition that X is xi, the probability of all possible values of yj is 1)

significance

Dimensionality reduction (or reduction of sample space)

Normalize using the marginal distribution law

application

throwing craps

Sample space: 32 sample points (i,j)

two-dimensional random variable

Continuous two-dimensional random vector

definition

The (X,Y) distribution function can be expressed as a variable upper limit integral function of a non-negative binary function on the plane domain.

Joint density function of (X,Y)

Nature (necessary and sufficient conditions)

non-negativity

normative

The probability value of a two-dimensional random variable falling in a certain area (finding the probability in the area -> finding the integral in the corresponding area)

(X,Y) edge density function about X and about Y

Derivation

Meaning (fix X or Y, no restriction on Y or X (that is, the integral limit is from negative infinity to positive infinity))

(X,Y) conditional probability density function

definition

First find the distribution function (limit integral mean value theorem) and find the derivative

Joint density function f(x,y)/marginal density function about Y (condition Y=y)

Nature (necessary and sufficient conditions)

non-negativity

normative

Under the condition that X is xi, the probability of all possible values of yj is 1

Geometric meaning (mnemonic)

Dimensionality reduction (or reduction of sample space)

Normalize using edge density function

Notice

The conditional density function can only be used when the condition is X=x (Y=y)

The condition is X≤x (a certain area), etc. You need to directly use the definition of conditional probability (less test)

Mutual conversion of (X,Y) joint density and conditional probability density

multiplication formula

application

form a triangle

Notice

For continuous random variables, the probability of an isolated point or curve is 0.

At the continuous point of f(x,y), the second-order mixed partial derivative of the joint distribution function is equal to the joint density function

Section 4 Distribution of two random variable functions Z=g(X,Y)

X and Y are both discrete

1. All possible values of Z Z=g(X,Y)

2. For each value of Z, all possible values of X and Y corresponding to it, sum the corresponding probabilities pij

X and Y are both continuous

Z=X Y(X,Y are independent)

calculate

Convolution formula

I. f(x,y)>0 determines the change range of the two independent variables {x,z-x} or {z-y,y}

II. The range of independent variable changes--Drawing->Determine the support area (f(x,z-x) or f(z-y,y)>0)

III. Fix z, and integrate x(y) from -∞ to ∞ (discuss the range of z points)

Definition

I. Find the joint density function f(x,y) of X and Y

II. Definition method -> Classification discussion scope (the integral area must be in the supporting part to be meaningful)

III. The distribution function is derived to obtain the density function

If it is not troublesome to determine the limit, it is simpler to use the convolution formula. If it is difficult to determine the limit, it is better to use the definition method.

example

Basics P240

Maximum and minimum value distribution

Z=max{X,Y}

Z=min{X,Y}

Opposite Events Act

and event addition formula

Generalization (n independent and identically distributed random variables)

X is continuous, Y is discrete

Section 3 Two-dimensional uniform distribution and two-dimensional normal distribution

normal distribution over area

definition

Wait for the possibility

Calculation (ratio of area)

One-dimensional normal distribution

Look-up table, normalization, symmetry, fixed parameters

Two-dimensional normal distribution

nature

Two-dimensional normal X, Y independent <---> correlation coefficient is 0

Two-dimensional normal --->X, Y normal; X, Y normal --/-> two-dimensional normal

Rotation (determinant ≠0)

Section 2 Independence of random variables (can be disassembled)

definition

Joint distribution function = product of marginal distribution functions (continuous)

Joint distribution law = product of marginal distribution laws (discrete)

X, Y independent <-> density function expressed as the product of two edge density functions (separably)

separated form

Determine whether the numerical part of the non-zero part of f(x,y) and the definition area (support area) are separable

Are the support areas separable? Taste slowly