# 110 年 - 110 國立高雄大學_碩士班招生考試_金融管理學系：統計學#102158

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1.

1. Given n observed data points, say for i = 1,2,3,..., n , please answer the following questions:

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(1). (10%) Please show that the straight line best fits the points is , where 【非選題】
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(2). (3%) Given five observed data points in Table 1: Please use the results of (1) to find the least-squares line (regression line).

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(3). (5%) Please evaluate the coefficient of determination 【非選題】
4.

2. Let , for i = 1, 2,3,... , be independent and identically distributed (i.i.d.) random variables. The expectation and variance of each are μ and σ2 , respectively. Please answer the following questions:

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(1). (5%) What is the probability distribution of ?

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(2). (10%) Let x1 , x2 , x3 , …, xm be m samples of a random variable X . If m is sufficiently large ( m  30 ), please show that the 95% confidence interval for the mean of X is: where and denote the sample mean and sample standard deviation, respectively.

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6.3. (10%) A teacher believes that the average grade of a class of 100 students is not higher than 85. The teacher randomly selects 64 students. The average grade of the 64 students is 87.8 and the standard deviation is 12.8. Using a significance level of 0.05, what conclusion should the teacher make?

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7.

4. (5%) Let and Var ( X ) denote the expectation and variance of a random variable X , respectively. Please show that 【非選題】
8.

5. Let X be normally distributed with mean 0 and variance 1. The probability density function of X is . Please answer the following question:

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(1). (7%) What is the probability density function of , where μ and σ>0 are constant?

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9.【題組】(2). (7%) What is the probability density function of X2 ?

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10.

≤≤≤≤6. Let N(t) , where 0 ≤ t ≤ T , denote a stochastic process, which satisfies the following conditions:
i. N (0)= 0 .
ii. N (t2)- N (t1) is normally distributed with mean 0 and variance t2 - t1 , where
0 ≤ t1 < t2 ≤ T .

iii. N(t) has independent increments. That is, given  are mutually independent.

【題組】 (1). (7%) Please evaluate the expectation of N(s)N(t) , where 0≤ s < t ≤ T . ( Hint: N(t)-N(s))+N(s)-N(0)))

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(2). (7%) Please show that the correlation coefficient between N(s) and N(t) is , where 0≤ s < t ≤ T .

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(3). (12%) Let denote the price of a stock, where S ( 0 )= 10 is the share price at t = 0 . What is the expected share price at t = T ? That is, please evaluate the expectation of S(T) . ( Hint: N(t)=N(t)-N(0) is normally distributed.)

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(4). (12%) Let X = . Please show that the expectation of X is You may begin with finding the minimum value of N(1) such that S(1)≥ 10 .