108年 - 108 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#147249

科目：研究所、轉學考(插大)◆機率 | 年份：108年 | 選擇題數：0 | 申論題數：16

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所屬科目：研究所、轉學考(插大)◆機率

選擇題 (0)

申論題 (16)

1. An urn contains four $2 bills and one $5 bill. Let the random variable X be the total amount that results when two bills are drawn from the urn with replacement (meaning that a bill is firstly drawn and put back, and then the other bill is drawn again). (totally 20%)

(a) Find the pmf of X. (10%)
(b) Find the expected value of X. (5%)
(c) Find the variance of X. (5%)
2. Let \( X \) be the number of successful events in a sequence of \( N \) events. Each event is independent and has the probability of success "p". Find the probability mass function (pmf). (10%)
3. Let \( X \) be the first wrong bit in a sent bit sequence. Then \( X \) is a geometric random variable with the parameter \( p \) (meaning that the probability of sending a wrong bit is \( p \) and the event of sending a bit is independent). If we know the first 4 bits are correct, please find the probability that the first 10 bits are correct. (10%)
4. A number \( \mathbf{N} \) of people entering the front gate of NTPU is a Poisson random variable. Assume that the average people entering the front gate within one hour are 50. (totally \( 10\% \))

(Just write down the expressions for answers. No need to calculate the values.)

(a) Please write down the pmf of N. (5%)
(b) Find the probability that the first person shows up after \(0.5\mathrm{hr}\) \((5\%)\)
5. Suppose that the random variable $X$ has the c.d.f.

$$F(x) = \frac{x^2}{16}, 0 \le x \le 4.$$

Determine (6% each, totally 24%)
(a) the p.d.f. of $X$;
(b) the mean of $X$;
(c) the variance of $X$; and
(d) the median of $X$.
6. Suppose that three random variables $X_1, X_2$ and $X_3$ are jointly distributed with the following p.d.f.

$$f(x_1, x_2, x_3) = 8e^{-2x_1 - kx_2 - x_3}$$

for $x_1 > 0$, $x_2 > 0$, and $x_3 > 0$. Determine (2% for (a), and 6% for (b)-(e), totally 26%)

(a) the value of the constant $k$;
(b) the marginal joint p.d.f. for $X_1$ and $X_3$;
(c) the probability $\Pr(X_1 < 1 | X_2 = 2, X_3 = 3)$;
(d) the sum of conditional expectations $E[X_1 | X_2] + E[X_2 | X_1]$;
(e) the covariance $\text{Cov}(X_1, X_2)$.