The authors of the article "A Probabilistic Insulation Life Model for Combined Thermal-Electrical Stresses" (IEEE Trans. on Elect. Insulation, 1985:519--522) state that "the Weibull distribution is widely used in statistical problems relating to aging of solid insulating materials(固体绝缘材料) subjected to aging and stress." They propose the use of the distribution as a model for time (in hours) to failure of solid insulating specimens(样本) subjected to AC voltage(交流电压). The values of the parameters depend on the voltage and temperature; suppose $\alpha=2.5$ and $\beta=200$ (values suggested by data in the article).

• (a) What is the probability that a specimen's lifetime is at most $200$? Less than $200$? More than $300$?
• (b) What is the probability that a specimen's lifetime is between $100$ and $200$?
• (c) What value is such that exactly $50\%$ of all specimens have lifetimes exceeding the value?
   

Weibull 分布是由瑞典物理学家 Waloddi Weibull 于1939年引进的.

他的1951年的一篇文章 "A Statistical Distribution Function of Wide Applicability"(J. Applied Mechanics, vol. 18: 293--297) 中讨论了一些应用.

定义: 

A random variable $X$ is said to have a Weilbull distribution with parameters $\alpha$ and $\beta$ ($\alpha > 0$, $\beta > 0$) if the pdf of $X$ is

\[
f(x;\alpha,\beta)=\begin{cases}
\frac{\alpha}{\beta^{\alpha}}x^{\alpha-1}e^{-(\frac{x}{\beta})^{\alpha}}, & x\geqslant 0,\\
0, & x < 0
\end{cases}
\]

与之十分相近的是 $\Gamma$-分布(Gamma distribution). 回顾其 pdf 定义为

\[
f_2(x;\alpha,\beta)=\begin{cases}
\frac{1}{\beta^{\alpha}\Gamma{\alpha}}x^{\alpha-1}e^{-\frac{x}{\beta}}, & x\geqslant 0,\\
0, & x < 0
\end{cases}
\]

令 Weibull 分布和 $\Gamma$-分布中的参数 $\alpha=1$, $\lambda=\frac{1}{\beta}$, 我们都可以得到指数分布(exponential distribution)

\[
f(x;\lambda)=\begin{cases}
\lambda e^{-\lambda x}, & x\geqslant 0,\\
0, & x < 0.
\end{cases}
\]

也就是说, 指数分布同是 $\Gamma$-分布和 Weibull 分布的特殊情形. 但是, 存在 $\Gamma$-分布, 其不属于 Weibull 分布; 也存在 Weibull 分布, 不属于 $\Gamma$-分布.

对于所定义的 Weibull 分布, 证明其 cdf 为

\[
F(x;\alpha,\beta)=\begin{cases}
1-e^{-(\frac{x}{\beta})^{\alpha}}, & x\geqslant 0,\\
0, & x < 0.\\
\end{cases}
\]

Prop. 若 $X$ 是服从 Weibull 分布的随机变量, 则其均值和方差为

\[
\begin{eqnarray}
\mu=E(X)&=&\beta\cdot\Gamma(1+\frac{1}{\alpha})\\
\sigma^2=V(X)&=&\beta^2\cdot\biggl[\Gamma(1+\frac{2}{\alpha})-\Bigl(\Gamma(1+\frac{1}{\alpha})\Bigr)^2\biggr]
\end{eqnarray}
\]

Prop. Let $X$ have a gamma distribution with parameter $\alpha$ and $\beta$. Then for any $x > 0$, the cdf of $X$ is given by

\[P(X\leqslant x)=F(x;\alpha,\beta)=F(\frac{x}{\beta};\alpha)\]

where $F(\cdot;\alpha)$ is the incomplete gamma function.

Let $X$ denote the distance (m) that an animal moves from its birth site to the first territorial vacancy it encounters(從出生地移動到它遇到的第一個領土空缺). Suppose that for banner-tailed kangaroo rats, $X$ has an exponential distribution with parameter $\lambda=.01386$ (as suggested in the article "Competition and Dispersal from Multiple Nests," Ecology, 1997:873--883).

• (a) What is the probability that the distance is at most $100$ m? At most $200$ m? Between $100$ and $200$ m?
• (b) What is the probability that distance exceeds the mean distance by more than $2$ standard deviation?
• (c) What is the value of the median distance?
   

Suppose that when a transistor(晶体管) of a certain type is subjected to an accelerated life test, the lifetime $X$ (in weeks) has a gamma distribution with mean $24$ weeks and standard deviation $12$ weeks.

• (a) What is the probability that a transistor will last between $12$ and $24$ weeks?
• (b) What is the probability that a transistor will last at most $24$ weeks? Is the median of the lifetime distribution less than $24$? Why or why not?
• (c) What is the $99$th percentile of the lifetime distribution?
• (d) Suppose the test will actually be terminated after $t$ weeks. What value of $t$ is such that only $.5\%$ of all transistors would still be operating at termination?
   

Suppose the time (in hours) taken by a homeowner to mow his lawn(修剪他的草坪) in an rv $X$ having a gamma distribution with parameters $\alpha=2$ and $\beta=\frac{1}{2}$. What is the probability that it takes:

• (a) At most $1$ hour to mow the lawn?
• (b) At least $2$ hours to mow the lawn?
• (c) Between $.5$ and $1.5$ hours to mow the lawn?
   

Evaluate the following:

• (a) $\Gamma(6)$
• (b) $\Gamma(\frac{5}{2})$
• (c) $F(4;5)$  (the incomplete gamma function)
• (d) $F(5;4)$
• (e) $F(0;4)$
   

Suppose the diameter at breast height(in.) of trees of a certain type is normally distributed with $\mu=8.8$ and $\sigma=2.8$, as suggested in the article "Simulating a Harvester-Forwarder Softwood Thinning" (Forest Products J., May 1997:36--41).

• (a) What is the probability that the diameter of a randomly selected tree will be at least $10$ in.? Will exceed $10$ in.?
• (b) What is the probability that the diameter of a randomly selected tree will exceed $20$ in.?
• (c) What is the probability that the diameter of a randomly selected tree will be between $5$ and $10$ in.?
• What value $c$ is such that the interval $(8.8-c,8.8+c)$ includes $98\%$ of all diameter values?
   

Remark:

For the definition of DBH(Diameter at Breast Height), we can refer to https://www.thoughtco.com/what-is-diameter-breast-height-1341720 .

Suppose the force acting on a column that helps to support a building is normally distributed with mean $15.0$ kips and standard deviation $1.25$ kips. What is the probability that the force

• (a) Is at most $17$ kips?
• (b) Is between $10$ and $12$ kips?
• (c) Differs from $15.0$ kips by at most $2$ standard deviations?
   

Remark:

1kips=1千磅=1000磅=453.59237千克(kg)

假设有助于支撑建筑物的作用在柱上的力满足均值为 15千磅, 标准差为 1.25千磅的正态分布.

求

• (a) 力至多为 $17$ 千磅的概率是多少?
• (b) 力在 $10$ 千磅和 $12$ 千磅之间的概率是多少?
• (c) 力和 $15$ 千磅之差在 $2$ 个标准差之间的概率是多少?

Determine $z_{\alpha}$ for the following:

• (a) $\alpha=.0055$
• (b) $\alpha=.09$
• (c) $\alpha=.663$
   

Here $z_{\alpha}$ denote the value on the measurement axis for which $\alpha$ of the area under the $z$ curve lies to the right of $z_{\alpha}$.