Questions in category: 概率论 (Probability)
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21. [Prop] $F(x;\alpha,\beta)=F(\frac{x}{\beta};\alpha)$

Posted by haifeng on 2020-05-11 10:32:32 last update 2020-05-11 10:32:32 | Answers (1) | 收藏


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.

22. [Exer12-4] Exercise 59 of Book {Devore2017B} P.177

Posted by haifeng on 2020-05-08 09:06:54 last update 2020-05-11 11:03:25 | Answers (1) | 收藏


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?
     

 

23. [Exer12-3] Exercise 57 of Book {Devore2017B} P.177

Posted by haifeng on 2020-05-08 09:05:52 last update 2020-05-08 09:05:52 | Answers (1) | 收藏


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?
     

 

24. [Exer12-2] Exercise 55 of Book {Devore2017B} P.177

Posted by haifeng on 2020-05-08 09:03:56 last update 2020-05-10 22:13:31 | Answers (1) | 收藏


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?
     

 

25. [Exer12-1] Exercise 53 of Book {Devore2017B} P.177

Posted by haifeng on 2020-05-08 08:31:13 last update 2020-05-08 08:31:13 | Answers (1) | 收藏


Evaluate the following:

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

 

26. [Exer11-4] Exercise 33 of Book {Devore2017B} P.169

Posted by haifeng on 2020-04-29 17:24:13 last update 2020-04-29 19:58:50 | Answers (1) | 收藏


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 .

27. [Exer11-3] Exercise 31 of Book {Devore2017B} P.169

Posted by haifeng on 2020-04-29 17:23:10 last update 2020-04-29 19:29:37 | Answers (1) | 收藏


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.

  • (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?

 

28. [Exer11-2] Exercise 29 of Book {Devore2017B} P.169

Posted by haifeng on 2020-04-29 17:22:10 last update 2020-04-29 18:23:46 | Answers (1) | 收藏


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 $z_{\alpha}$.

29. [Exer11-1] Exercise 27 of Book {Devore2017B} P.169

Posted by haifeng on 2020-04-29 17:21:20 last update 2020-04-29 17:21:20 | Answers (1) | 收藏


Let $Z$ be a standard normal random variable.


In each case, determine the value of the constant $c$ that makes the probability statement correct.

  • (a) $\Phi(c)=.9838$
  • (b) $P(0\leqslant Z\leqslant c)=.291$
  • (c) $P(c\leqslant Z)=.121$
  • (d) $P(-c\leqslant Z\leqslant c)=.668$
  • (e) $P(c\leqslant|Z|)=.016$
     

 

30. [Exer10-4] Exercise 19 of Book {Devore2017B} P.157

Posted by haifeng on 2020-04-21 09:20:14 last update 2020-04-22 23:18:00 | Answers (1) | 收藏


Let $X$ be a continuous rv with cdf
\[
F(x)=\begin{cases}
       0, & x\leqslant 0, \\
       \frac{x}{4}\Bigl[1+\ln(\frac{4}{x})\Bigr], & 0 < x\leqslant 4, \\
       1, & x > 4.
     \end{cases}
\]
[This type of cdf is suggested in the article "Variability in Measured Bedload-Transport Rates" (Water Resources Bull., 1985:39--48) as a model for a certain hydrologic variable.] What is

  • (a) $P(X\leqslant 1)$?
  • (b) $P(1\leqslant X\leqslant 3)$?
  • (c) The pdf of $X$?
     

 

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