109年 - 109 國立臺北大學_碩士班一般入學考試試題_通訊工程學系：機率#147256

科目：研究所、轉學考(插大)◆機率 | 年份：109年 | 選擇題數：0 | 申論題數：20

試卷資訊

所屬科目：研究所、轉學考(插大)◆機率

選擇題 (0)

申論題 (20)

1. (5% each, 20% in total) Yes/No questions. No need to state reasons.

(a) Consider three events \( E_1, E_2 \), and \( E_3 \). Is the statement "if \( P[E_1 \cap E_2 \cap E_3] = P[E_1]P[E_2]P[E_3] \), then \( E_1, E_2 \), and \( E_3 \) are independent" true?
(b) Consider two events \( E_{1} \) and \( E_{2} \). Is the statement "if \( E_{1} \cap E_{2} = \emptyset \), then \( P[E_{1} \cup E_{2}] = P[E_{1}] + P[E_{2}] \)" true?
(c) Let \(\Omega\) be a sample space. Let \(E_1\) and \(E_2\) be two disjoint events, with \(E_1 \cup E_2 = \Omega\). Is the statement " \(P[E] = P[E \cap E_1] + P[E \cap E_2]\) for any event \(E \subseteq \Omega\) true?
(d) Consider a random variable \( X \). Let random variable \( Y = a \cdot X + b \), for some constants \( a \) and \( b \). Let \( Var[X] \) and \( Var[Y] \) be the variances of \( X \) and \( Y \), respectively. Is the statement "Var[Y] = a · Var[X] + b" true?
2. (5% each, 15% in total) Consider a random variable $X$ with the PMF

$$P_X(x) = \begin{cases} c & \text{if } x = 1; \\ \binom{1}{2}^x & \text{if } x = 2, 3, 4 \cdots; \\ 0 & \text{else}, \end{cases}$$

for some constant $c$.

(a) What is the constant \( c \)?
(b) What is the probability \( P[X \in \{0,1\}] \)?
(c) What is the mean \( E[X] \)?
3. (5% each, 15% in total) Let $X$ be the number of accesses at a website in a day. Suppose that $X$ is the Poisson random variable with mean $\alpha$.

(a) What is the probability that no one accesses the website in that day?
(b) What is the probability that no one accesses the website during 10am and 11am in that day?
(c) Suppose that each access at the website consumes a power of \( c \) units, i.e., \( c \cdot X \) units of power consumption in that day. What is the expected power consumption in that day?
4. (2% for (a), and 4% for (b)-(e), totally 18%) Let $X$ be a random variable with the following probability density function (p.d.f.)

$$f_X(x) = \begin{cases} cx, & 0 \le x \le 2; \\ 2c(3-x), & 2 < x \le 3; \\ 0, & \text{otherwise}. \end{cases}$$

where $c$ is a constant.

(a) Find the value of \( c \).
(b) Find the mean of \( X \).
(c) Find the variance of \( X \).
(d) Find the median of \( X \).
(e) Let \( Y = X^2 \). Find the mean of \( Y \).
5. (6% for each, totally 24%) Let $X$ and $Y$ be two independent random variables with identical uniform distribution over $[0,2]$. Let $Z = |Y - X|$.

(a) Find the probability \( \mathrm{P}(X < 2Y) \).
(b) Find the conditional probability \( \mathrm{P}(2X < Y|X < 2Y) \).
(c) Find the cumulative distribution function (c.d.f.) of \( Z \).
(d) Find the p.d.f. of \( Z \).
6. (8%) Let $X$ and $Y$ be two random variables. $X$ has the p.d.f. $f_X(x) = 2x, 0 \le x \le 1$, and $Y = g(X)$. If $Y$ has the p.d.f. $f_Y(y) = e^{-y}, y > 0$, find the function $g(x)$.