107年 - 107 國立嘉義大學_碩士班(乙組)招生考試試題_應用數學系：機率統計#146100

1. A biased coin is tossed till a head appears for the first time. What is the probability that the number of required tossed is odd? (10 points)

2. Show that \( P(AB|C) = P(A|BC) P(B|C) \). (10 points)

(a) Find P(4). (5 points)

(b) Find \( \mu \) and \( \sigma^2 \). (5 points)

(c) What is the probability that x is greater than 2? (5 points)

4. Let $X$ be a random variable and let $g(x)$ be a nonnegative function. For any $r > 0$, then show that $P(g(x) \ge r) \le \frac{E(g(x))}{r}$. (15 points)

\( Z_{0.05}=1.65, Z_{0.025}=1.96, \chi_{0.05,5}^{2}=11.07, \chi_{0.05,6}^{2}=12.59, \chi_{0.05,7}^{2}=14.07 \)

1. A dice is thrown 90 times. Different faces occurred 11, 17, 20, 15, 13, 14 times, respectively. Test whether the dice is balanced. (10 points)

2. Two random samples of 8 and 11 plots are chosen independently from two distant places. The yields of maize obtained from these plots are given below. Test whether the average maize yield is same for two places. (15 points)
First place: 12, 15, 16, 11, 13, 13, 14, 15
Second place: 15, 22, 12, 25, 18, 19, 20, 17, 25, 24, 18

(a) Level of significance means rejecting a true null hypothesis.

(b) In hypothesis testing, error is always committed.

(c) Accepting a false null hypothesis is type II error.

(d) In hypothesis testing type I error is always committed.

(a) Calculate correlation coefficient between x and y. (5 points)

(b) Test whether the population correlation coefficient is significant. (5 points)

(c) Test whether the population correlation coefficient is greater than 0.5. (5 points)