111年 - 111 國立臺北大學_日間學士班轉學生招生考試試題_統計學系日間學士班3年級：機率概論#147608

科目：【阿摩】未分類題庫 | 年份：111年 | 選擇題數：0 | 申論題數：19

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所屬科目：【阿摩】未分類題庫

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申論題 (19)

1. (10%) Let $C_1$, $C_2$ and $C_3$ be mutually independent events, where $P[C_1] = \frac{2}{6}$, $P[C_2] = P[C_3] = \frac{1}{6}$. Find $P[C_1 \cup C_2 \cup C_3]$ and $P[(C_1^C \cap C_2^C) \cup C_3]$, where $C^C$ is the complement of $C$.
2. (20%) An urn contains five balls, one marked $W$ and four marked $L$. You and another player take turns selecting a ball at random from the urn, one at a time. The first person to select $W$ ball is the winner.

(a) (5%) Write down the sample space if balls are selected without replacement.
(b) (5%) Continued (a), find the probability that you are the winner at third draw.
(c) (5%) If balls are selected with replacement. Let \( X \) denote the number of selected balls if you are the winner. Find the probability mass function.
(d) (5%) Derive \( E[X] \). [A detailed derivation is required.]
3. (10%) Let $X$ have the probability mass function

$$f_X(x) = P[X = x] = \frac{3}{2} \left(\frac{1}{2}\right)^{2|x|}, x = \pm 1, \pm 2, \dots$$

(a) \((5\%)\) Find \(P[X = 1,2,3,\dots ]\)
(b) (5%) Find \( E[X] \) and \( Var(X) \). [A detailed derivation is required and the answer has to be simplified.]
4. (10%) Suppose that 500 points are selected independently and at random from the unit square

$$\{(x, y): 0 \le x < 1, 0 \le y < 1\}.$$

Let $W$ equal the number of points that fall into $\{(x, y): x^2 + y^2 < 1\}$.

(a) (5%) Write down the distribution of \( W \) and the corresponding probability mass function.
(b) (5%) Find the mean and variance of \( W \). [A detailed derivation is required.]
5. (15%) Red Rose tea randomly began placing 1 of 6 English porcelain miniature figurines in a 100-bag box of the tea, selecting from 6 nautical figurines, which are numbered 1 to 6.

(a) Let \( X \) denote the number of Red Rose tea that a family would have to buy to have at least three of the nautical figurine having number 1. Write down the probability mass function of \( X \).
(b) Derive the moment generating function of \( X \). [A detailed derivation is required.]
(c) On the average, how many boxes of tea must be purchased by a customer to obtain a complete collection consisting of the 6 nautical figurines.
6. (10%) Let $f(x) = \frac{1}{6}, 0 \le y \le 4, y \le x \le y + 2$ be the joint pdf of $X$ and $Y$.

(a) Find the conditional pdf of \( X \) given \( Y = y \) and of \( Y \) given \( X = x \).
(b) Find the conditional expectation of \( X \) given \( Y = y \).
7. (5%) Let $X_1$ and $X_2$ have the joint probability density function

$$f(x_1, x_2) = \frac{1}{2}, 0 < x_1 < x_2 < 2$$

Let $Y = \frac{X_1}{X_2}$. Find the probability density of $Y$.
8. (10%) Two components operate in parallel in a device. The device fails when and only when both components fail. The lifetimes, $X_1$ and $X_2$, of the respective components are independent and identically distributed with an exponential distribution with the probability density function.

The cost of operating the device is $Z = Y_1 + 2Y_2$, where $Y_1 = \min(X_1, X_2)$ and $Y_2 = \max(X_1, X_2)$

(a) Find the joint probability density function of \(Y_{1}\) and \(Y_{2}\).
(b) Find E[Z].
9. (10%) In a college health fitness program, let $X$ denote the weight in kilograms of a male freshman at the beginning of the program and $Y$ denote the weight change during a semester. Assume $X$ and $Y$ have a bivariate normal distribution with $\mu_X = 75$ and $\mu_Y = 5$, $\sigma_X^2 = 64$, $\sigma_Y^2 = 2.56$ and $\rho = -0.6$. The joint probability density function of a bivariate normal distribution is defined as

$$f(x, y) = \frac{1}{2\pi\sigma_X\sigma_Y\sqrt{1-\rho^2}} \exp\left(-\frac{q(x, y)}{2}\right)$$

where $q(x, y) = \frac{1}{1-\rho^2} \left[ \left(\frac{x-\mu_X}{\sigma_X}\right)^2 - 2\rho \left(\frac{x-\mu_X}{\sigma_X}\right) \left(\frac{y-\mu_Y}{\sigma_Y}\right) + \left(\frac{x-\mu_Y}{\sigma_Y}\right)^2 \right]$.

(a) Derive the marginal pdf of \( Y \) and find \( P[2.5 \leq Y \leq 6.5] \). [A detailed derivation is required.]
(b) Derive the conditional pdf of \( Y \) given \( X = x \) and find

$P[2.5 \leq Y \leq 6.5 | X = 80]$. [A detailed derivation is required.]