108 年 國立成功大學自然災害減災及管理國際碩士學位學程《統計學》

A random experiment has a sample space $S = \{a, b, c, d\}$. These outcomes are not equally-likely; their probabilities are: $P(a) = 0.1$, $P(b) = 0.3$, $P(c) = 0.5$, $P(d) = 0.1$. Let Event $A = \{a, b\}$, Event $B = \{b, c, d\}$, and Event $C = \{d\}$. $A'$ is the complement of event $A$. Determine the following probabilities:
(a) $P(A)$
(b) $P(B)$
(c) $P(A')$
(d) $P(A \cap B)$
(e) $P(A \cup B)$

For the case of the thin copper wire, suppose that the number of flaws follows a Poisson distribution (the probability density function is $f(x) = \frac{e^{-\lambda}\lambda^x}{x!}$) with a mean of 2.3 flaws per mm.
(a) Find the probability of exactly 2 flaws in 1 mm of wire.
(b) Determine the probability of 10 flaws in 5 mm of wire.
(c) Determine the probability of at least 1 flaw in 2 mm of wire.

Suppose that the current measurements in a strip of wire are assumed to follow a Normal distribution with $\mu = 10$ and $\sigma = 2$ mA.
(a) What is the probability that the current measurement is between 9 and 11 mA?
(b) Determine the value for which the probability that a current measurement is below 0.98.
[Hint: Assume $Z$ is a standard normal random variable. $P(Z \le -0.5)=0.30854$; $P(Z \le 0.5)=0.69146$; $P(Z \le 1.5)=0.93319$; $P(Z \le 2.05)=0.98000$]

Use the least-square method to derive the slope of a simple linear regression model $y = \beta_0 + \beta_1 x$ is $\beta_1 = \frac{\sum_{i=1}^{n}x_i y_i - n\bar{x}\bar{y}}{\sum_{i=1}^{n}x_i^2 - n\bar{x}^2}$, where $(x_i, y_i)$ is the $n$ paired data. $\bar{x}$ and $\bar{y}$ are the mean values of $x$ and $y$, and write the definition of coefficient of determination.

Use the Method of Moment to derive the parameters of Gamma distribution.
[Hint: PDF of Gamma distribution $f(x) = \frac{\lambda^r}{\Gamma(r)} x^{r-1} e^{-\lambda x}$ for $x>0$. And the first and second moments of Gamma distribution are $E[X] = \frac{r}{\lambda}$ and $E[X^2] = \frac{r(r+1)}{\lambda^2}$ respectively.]