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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
> 申論題
(b) Find the expected value of X. (5%)
相關申論題
(c) Find the variance of X. (5%)
#605315
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%)
#605316
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%)
#605317
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%)
#605318
(b) Find the probability that the first person shows up after \(0.5\mathrm{hr}\) \((5\%)\)
#605319
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 $X$;
#605320
(b) the mean of $X$;
#605321
(c) the variance of $X$; and
#605322
(d) the median of $X$.
#605323
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 $k$;
#605324
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