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

(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%)

(a) Please write down the pmf of N. (5%)

(b) Find the probability that the first person shows up after \( 0.5\mathrm{hr} \) \( (5\%) \)

(a) the p.d.f. of X;

(b) the mean of X;

(c) the variance of X; and

(d) the median of X.

(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 $\text{Cov}(X_1, X_2)$.