12. For positive random variables X and Y , suppose E (Y / X ) = θ X , which of the following is false ?
(A) If a random variable , then E ( Z ) = θ .
(B) The estimator is unbiased for θ .
(C) The estimator is biased for θ .
(D) W₂ is not the same as W₁ .

1. x1 , x2 , x3 ,...., xn are randomly samples from a normal population N ( μ , σ2 ) . If we use  to estimate σ2 , then = (A) (B)σ2 (C) (D)

2. A continuous random variable X has the following probability density function: f ( x) = , 0 ≤ x ≤ 2 . If we have another other random variable Y = 3 X2 + 2 , the probability density function of Y is (A) (B) (C)f(y)=3(4y-1)2+2 (D)

3. Following 2., Pr( y ≤ 5 )= (A) (B) (C) (D)

4. When the distribution of a random variable is symmetric, which of the following parameters is zero? (A) mean (B) variance (C) third moment (D) fourth moment

5. Which of the following statements is true? (A) In a simple regression model, the determinant is zero means there is no relation between the dependent and independent variables. (B) If the distribution of the disturbance term is not a normal distribution, then the OLS estimator cannot be unbias. (C) If the intercept of the regression function is very close to zero, we can infer that the intercept is not significantly different from zero. (D) If we use t-distribution to test whether the coefficient of the regression function is significantly differently from zero, the distribution of the OLS estimator of the coefficient should be a normal distribution.

6. The probability density function of discrete random variable x is : , the which of the followings is not true ? (A) p lim x = 1 . (B) (C) (D) x will converge to a given vale.

7. n samples are drew from a normal distribution N ( μ , σ2 ) , = (A)σ2 (B)μ2(C)σ2+μ2 (D)

8. X , Y , Z are random variables, which of the following statement about conditional means is not true? (A) E[ E (Y / X )] = E (Y ) . (B) If X and Y are independent, then E (Y / X ) = E (Y ) . (C) E (Y / X ) = E[ E (Y / X , Z ) / X ] . (D) If Cov( X , Y ) = 0 , then E (Y / X ) = E (Y ) must hold.

9. Which of the following results of sampling rejects the null hypothesis that the population mean is significantly different from than 0. (A) p-value is very low. (B) t-value is positive. (C) The confidence interval of the sample mean includes 0. (D) t-value is negative and close to zero.

複選題10. Which of the following statement is true? (A) In hypothesis testing, if we fail in rejecting the null hypothesis when it is actually false, then we make a Type I error. (B) When we have the same sample to test the mean of a population around a specific value, it is more likely to reject the null hypothesis when we use one-tailed test than when we use two-tailed test under the same significant level. (C) The probability of t-value less than that of the critical t-value of a significance level is equal to p-value. (D) Using confidence interval to see whether it includes the specific population parameter may make different inference from doing hypothesis testing.

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