[PubMed Mathematics] Probability Theory and Mathematical Statistics - Chapter 2 | One-dimensional Random Variables and Their Distributions (2, Common Random Variables and Their Distributions | Distribution of Random Variable Functions)

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introduction

Continuing from the previous article, we continue to study the second part of Chapter Two, the second part of one-dimensional random variables and their distribution.

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3. Common random variables and their distributions

3.1 Common discrete random variables and their distribution laws

(a) (0-1) distribution

Let random variable $The\; possible\; value\; of\; X$ is 0 or 1, and its probability is$P$ { $X=1$ } $=p,$ $P$ { $X=0$ } $=1-p(0<p<1$ , called$X$ obeys (0-1) distribution, denoted as$X\sim B(1,p).$

(2) Binomial distribution

Let random variable $The\; distribution\; law\; of\; X$ is$P$ { $X=k$ } $=C_n^k{p}^k(1-p{)}^{n-k}$ ，其中 $k=0,1,2,\dots ,n,0<p<1,$ called random variable$X$ obeys the binomial distribution, denoted as$X\sim B(n,p).$

Recall the Bernoulli probability in Chapter 1, which is also a binomial distribution.

(3) Poisson distribution

Let the discrete random variable $Let\; X$ set P { X = k } = λ kk ! and − λ , P\{X=k\}=\frac{\lambda^k}{k!}e^{-\lambda},P{ X=k}=k!lk​e− λ ,among which，$l>0,k=0,1,2,\dots ,n,$ called random variable$X$ obeys the parameter$The\; Poisson\; distribution\; of\; \lambda$ , denoted as$X\sim P\left(\lambda \right).$

(4) Geometric distribution

Let the discrete random variable XXThe distribution law of$X$$$P\{X=k\}=p(1-p{)}^{k-1},$$where,$k=1,2,\dots ,n,$ called random variable$X$ obeys the geometric distribution, denoted as$X\sim G(p).$

Random variable XX subject to geometric distribution$X$ can be understood in this way: Suppose there are only two results$A,\overline{A},P(A)=p$ , then$X$ means AAin Bernoulli trialThe number of trials when$A\; first\; occurs.$
For example,$X=2$ , it means that the first test happened after two tests, that is, the first test did not happen, but the second test happened;$X=n$ , means the first$n-1$ trial did not occur,$n$ trials occur. This is easy to understand, and the formula can be remembered at once.

(5) Hypergeometric distribution

Let the discrete random variable XXThe distribution law of$X$ is P { X = k } = CM k ⋅ CN − M n − k CN n , P\{X=k\}=\frac{C_M^k \cdot C_{NM}^{nk}} {C_N^n},P{ X=k}=CNn​CMk​⋅CN−Mn−k​​, of which$M,N,k,n$ is a natural number, andM ≤ N , max { N − M , 0 } ≤ k ≤ min { M , n } , n ≤ NM \leq N,max\{NM,0\} \leq k \leq min\{ M,n\},n \leq NM≤N,max{ N−M,0}≤k≤min{ M,n},n≤N , called random variable$X$ obeys the hypergeometric distribution, denoted as$X\sim H(n,M,N).$

3.2 Common continuous random variables and their probability densities

(1) Uniform distribution

Let random variable XXThe probability density of$X$$$f(x)=\{\begin{array}{ll}\frac{1}{b-a},& a\le x\le b\\ 0,& else\end{array},$$ called random variable$X$ is in the interval$(a,b)$ The interior obeys a uniform distribution, which is denoted as$X\sim U(a,b).$

If the random variable $X\sim U(a,b)$ , then its distribution function is$$F(x)=\{\begin{array}{ll}0,& x<a\\ \frac{x-a}{b-a},& a\le x\le b\\ 1,& x\ge b\end{array}$$

(2) Exponential distribution

Let random variable XXThe probability density of$X$$$f(x)=\{\begin{array}{ll}\lambda e^{-\lambda x}& x>0\\ 0,& x\le 0\end{array}(l>0)$$ called random variable$X$ obeys the parameter$The\; exponential\; distribution\; of\; \lambda$ , denoted as$X\sim E\left(\lambda \right).$

If the random variable $X\sim E\left(\lambda \right)$ , then its distribution function is$$F(x)=\{\begin{array}{ll}1-e^{-\lambda x},& x\ge 0\\ 0,& x<0\end{array}$$

(3) Normal distribution

Let random variable $The\; X$ -variance function$$f(x)=\frac{1}{\sqrt{2p.m}p}{e}^{-\frac{(x-\mu {)}^2}{2p^2}}(-\infty <x<+\infty ),$$ called random variable$X$ obeys normal distribution, denoted as$X\sim N(\mu ,p^2)$, its probability density function is shown in the figure below:

especially, if$m=0,p=1$ , called random variable$X$ obeys the standard normal distribution, denoted as$X\sim N(0,1)$ , its probability density is$$\phi \left(x\right)=\frac{1}{\sqrt{2p.m}}{e}^{-\frac{x^2}{2}}(-\infty <x<+\infty ),$$ its probability density function is shown in the figure below:

the distribution function is$$\Phi \left(x\right)={\int }_{-\infty }^x\phi \left(t\right)dt.$$ The normal distribution has the following properties:

(1) If $X\sim N(0,1)$ , then its probability density function$\phi \left(x\right)$ is an even function, and P { X ≤ 0 } = Φ ( 0 ) = 0.5 , P\{X \leq 0 \}=\varPhi(0)=0.5,P{ X≤0}=Φ ( 0 )=0.5 ,$$P\{X\le -a\}=\Phi (-a)=P\{X>a\}=1-\Phi \left(a\right).$$ (2) If the random variable$X\sim N(\mu ,p^2)$，则P { X ≤ μ } = P { X > μ } = 0.5 , P\{X \leq \mu\}=P\{X > \mu\}=0.5,P{ X≤m }=P{ X>m }=0.5 , that is, the image of the density function of the normal distribution is about$x=\mu$ symmetry.

(3) If the random variable $X\sim N(\mu ,p^2)$, 则$\frac{X-}{p}\sim N(0,1).$

(4) If the random variable $X\sim N(\mu ,p^2)$，刪P { a < X ≤ b } = F ( b ) − F ( a ) = Φ ( b − μ σ ) − Φ ( a − μ σ ). P\{a < X \leq b\}=F(b)-F(a)=\varPhi(\frac{b-\mu}{\sigma})-\varPhi(\frac{a-\mu} {\sigma}).P{ a<X≤b}=F(b)−F(a)=F (pb−m)−F (pa−m). （5）$$\Phi \left(a\right)+\Phi \left(b\right)=\{\begin{array}{ll}<1,& a+b<0\\ =1,& a+b=0\\ >1,& a+b>0\end{array}$$

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4. Distribution of random variable function

Let $X$ is a random variable whose distribution is known, said$Y=\phi \left(X\right)$ is a random variable$function\; of\; X$ , study random variable$The\; distribution\; of\; Y$ and the distribution of a function of a random variable.

(1) Distribution of discrete random variable functions

Let $X$ is a random variable,$Y=\phi \left(X\right)$ , as long as according toXXThe possible value and probability of$X$ to find YYThe possible values ​​and probabilities of$Y$ can be obtained as YYThe distribution law of$Y.$

(2) Distribution of continuous random variable function

Let $X$ is a continuous random variable whose probability density is$f\left(x\right)$，又$Y=\phi \left(x\right)$ , find the random variable$For\; the\; distribution\; of\; Y$ , find$The\; Y$ function$$P\{AND\le y\}=P\left\{\phi \right(X)\le y\},$$ then pass$The\; distribution\; of\; X$ findsYYThe distribution of$Y.$