Abstract: 本文介绍Gamma分布相关知识的第二部分指数分布
Keywords: The Exponential Distributions

Gama分布

指数分布 The Exponential Distribution

Definition Exponential Distributions.Let $\beta >0$ .A random variable $X$ has the exponential distribution with parameter $\beta$ if $X$ has a continuous distribution with the p.d.f.
$$f(x\beta)= \begin{cases} \beta e^{-\beta x}& \text{ for }x>0\\ 0&\text{for} x\leq 0 \end{cases}$$

Gamma Distributions.Let $\alpha$ and $\beta$ be positive numbers.A random variable $X$ has the gamma distribution with parameters $\alpha$ and $\beta$ if $X$ has a continuous distribution for which the p.d.f. is
$$f(x|\alpha,\beta)= \begin{cases} \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}&\text{ for } x>0\\ 0&\text{otherwise} \end{cases}$$

Theorem The exponential distribution with parameter $\beta$ is the same as the gamma distribution with parameters $\alpha=1$ and $\beta$ .If $X$ has the exponential distribution with parameter $\beta$ ,then
$$E(X)=\frac{1}{\beta}\text{ and } Var(X)=\frac{1}{\beta^2}$$
and m.g.f. of $X$ is
$$\Psi(t)=\frac{\beta}{\beta-t}\text{ for } t<\beta$$

Theorem Memoryless Property of Exponential Distributions.Let $X$ have the exponential distribution with parameters $\beta$ ,and let $t>0$ .Then for every number $h>0$ ,
$$Pr(X\geq t+h|X\geq t)=Pr(X\geq h)$$

$$Pr(X\geq t)=\int^{\infty}_{t}\beta e^{-\beta x}dx=e^{-\beta t}\tag{5.7.19}$$

\begin{aligned} Pr(X\geq t+h|X\geq t)&=\frac{Pr(X\geq t+h)}{Pr(X\geq t)}\\ &=\frac{e^{-\beta(t+h)}}{e^{-\beta t}}=e^{-\beta h}=Pr(X\geq h) \end{aligned}\tag{5.7.20}

使用寿命测试 Life Tests

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Theorem Suppose that the variables $X_1,\dots,X_n$ form a random sample from the exponential distribution with parameter $\beta$ .Then the distribution of $Y_1=min{X_1,\dots,X_n}$ will be the exponential distribution with parameter $n\beta$

\begin{aligned} Pr(Y_1>t)&=Pr(X_1>t,\dots,X_n>t)\\ &=Pr(X_1>t)\dots Pr(X_n>t)\\ &=e^{-\beta t}\dots e^{-\beta t}=e^{-n\beta t} \end{aligned}

Theorem Suppose that the variables $X_1,\dots,X_n$ form a random sample from the exponential distribution with parameters $\beta$ .Let $Z_1\leq Z_2\dots \leq Z_n$ be the random variables $X_1,\dots,X_n$ sorted from smallest to largest.For each $k=2,\dots,n$ ,let $Y_k=Z_k-Z_{k-1}$ ,Then the distribution of $Y_k$ is the exponential distribution with parameter $(n+1-k)\beta$

指数分布和泊松过程的关系 Relation to Poisson Process

Theorem 5.7.12 Times between Arrivales in a Poisson Process.Suppose that arrivals occur according a Poisson process with rate $\beta$ .Let $Z_k$ be the time until the $k$ th arrival for $k=1,2,\dots$ .Define $Y_1=Z_1$ and $Y_k=Z_k-Z_{k-1}$ for $k\geq 2$ Then $Y_1,Y_2,\dots$ are i.i.d. and they each have the exponential distribution with parameter $\beta$

Corollary Time until $k$ th Arrival. In the situation of Theorem 5.7.12,the distribution of $Z_k$ is the gamma distribution with paramters $k$ and $\beta$

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