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18-19 (1.2.1 probability density)

2022-01-27 05:03:12 CNpupil

1.2.1 Probability density

   In addition to considering the probability defined on the discrete-time set , We also want to consider the probability of continuous variables . We will confine ourselves to relatively informal discussions . If the real valued variable x x ​ Fall in the range ( x , x + δ x ) (x,x+\delta x) ​ The probability within is determined by p ( x ) δ x p(x)\delta x Express δ x 0 \delta x\rightarrow 0 , that p ( x ) p(x) be called x x ​ Probability density on .​​ Pictured 1.12 Shown . x x In the interval ( a , b ) (a,b) The probability within is given by :

p ( x ( a , b ) ) = a b p ( x ) d x (1.24) p(x\in(a,b))=\int_a^bp(x)dx\tag{1.24}

Because of the probability yes Nonnegative , And because x x The value of must be somewhere on the real axis , So probability density p ( x ) p(x) These two conditions must be met

p ( x ) 0 (1.25) p(x)\geq0\tag{1.25}
p ( x ) d x = 1 (1.26) \int_{-\infty}^\infty p(x)dx=1\tag{1.26}

Under the nonlinear variation of variables , Due to the Jacobian factor , Probability density transformation is different from simple function . for example , If we consider variables x = g ( y ) x=g(y) The change of , So the function f ( x ) f(x) Turn into f ( y ) = f ( g ( y ) ) f(y)=f(g(y)) . Now consider the probability density p x ( x ) p_x(x) , It corresponds to the new variable y y The density of the p y ( y ) p_y(y) , Where enough means p x ( x ) p_x(x) and p y ( y ) p_y(y) It's a fact of different densities . For the smaller δ x \delta x Value , be located ( x , x + δ x ) (x,x+\delta x) Observations in the range will be converted to ( y , y + δ y ) (y,y+\delta y) Range , among p x ( x ) δ x p y ( y ) δ y p_x(x)\delta x \simeq p_y(y)\delta y ​, therefore

P ( z ) = z p ( x ) d x (1.28) P(z)=\int_{-\infty}^zp(x)dx\tag{1.28}

It meets the P ( x ) = p ( x ) P'(x)=p(x) ​​, Pictured 1.12 Shown .

Figure 1.12

chart 1.12 discrete The probability of the variable Can be diffused into continuous variables x x Probability density on p ( x ) p(x) , And make it in the interval ( x , δ x ) (x,\delta x) Of x x The probability of is determined by δ x \delta x Of p ( x ) δ x p(x)\delta x give δ x 0 \delta x\rightarrow0 . The probability density can be expressed as a cumulative distribution function P ( x ) P(x) The derivative of .

   If we have several continuous variables x 1 , . . . , x D x_1,...,x_D , With the vector x x Express , Then we can define the joint probability density p ( x ) = p ( x 1 , . . . , x D ) p(x)=p(x_1,...,x_D) bring x Fall into inclusion point x x Infinitesimal volume of δ x \delta x The probability in is determined by p ( x ) δ x p(x)\delta x give . This multivariate probability density must satisfy

p ( x ) 0 (1.29) p(x)\geq0\tag{1.29}
p ( x ) d x = 1 (1.30) \int p(x)dx=1\tag{1.30}

The integral is for the whole x x ​ Integral of space . We can also consider the combination of joint probability distribution on discrete variables and continuous variables .

   Be careful , If x x It's a discrete variable , that p ( x ) p(x) Sometimes referred to as the probabilistic mass function , Because it can be seen as a group focused on x x ​ At the allowable value “ Probability quality ”.

   Probability sum product rule and Bayesian theorem are also applicable to the case of probability density , Or a combination of discrete and continuous variables . for example , If x x and y y Are two real variables , Then the form of sum rule and product rule is

p ( x ) = p ( x , y ) d y (1.31) p(x)=\int p(x,y)dy\tag{1.31}
p ( x , y ) = p ( y x ) p ( x ) (1.32) p(x,y)=p(y|x)p(x)\tag{1.32}

The formal proof of the sum product rule of continuous variables requires a branch of mathematics called measurement theory , Not within the scope of this book . However , By dividing each real variable into width intervals , Its effectiveness can be seen informally Δ \Delta Consider the discrete probability distributions on these intervals . Walking limit Δ 0 \Delta\rightarrow0 ​ Then convert the sum to Integrate and give the desired result .

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