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

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.

(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.]

(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.]

(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.]

(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.

(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 $Y$.

(a) Find the joint probability density function of \( Y_{1} \) and \( Y_{2} \).

(b) Find E[Z].

(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.]