Question

3. [Multiple Choice Questions (4 points per question, 32 points in total): Please answer the following questions. 3.1 There is a random variable X with observations 3.5 Which of the following is implied from the law [X,,X2,...,X). It is known that these observations follow the normal distribution with mean μ and variance σ2. Which of the following will lead to a c? of large numbers? (a) If we have a sufficiently large number of observations, its sample mean is close to the population mean. (b) If we have a sufficiently large number of (b) (X - )/a2 (c) (X + μ)/ observations, its sample variance is close to the population variance. (c) If we have a sufficiently large number of observations, these observations follow a normal distribution. 3.2 In standard normal distribution, 99.7% of observations lie in the range between (d) None of the above. 3.6 In a simple regression model, y = + Ax + u which of the following is NOT referring to x? t variable (b) Explanatory variable (c) Control variable (d) Regressor (e) All of the above refer tox 3.3 A cumulative distribution function of a random variable X is by definition a probability that X will take a value (a) less than x (b) less than or equal to x (c) x (d) greater than x (e) greater than or equal to X 3.7 In a simple regression model, y-A, + Ax + u which of the following is NOT referring to y? t variable (b) Explained variable (c) Predicted variable (d) Regressant (e) All of the above refer to y 3.4 Fill in the blank to correctly describe the central limit theorem. Given a sample where its observations are followingdistribution, a newly created sample of "averages" from the original sample follows a normal distribution if we pick sub- samples many times and find "ave times (a) a Pareto (b) a normal (c) any (d) none of the above 3.8 Which of the following statements is correct? (a) If X and Y are independent, then E YX]- E[Y]. E[X] irrelevant from E[Y]. rages many (b) If X and Y are independent, then E YIX] (c) IfX and Y are independent, then E[YlX] is (d) None of the above is correct.

Answer 1

3.1) When mean = $\mu$ and variance = $\sigma^2$ then

$Z=\frac{X-\mu}{\sigma}$ which is called as standard normal distribution.

Correct Anwer :(a) $\frac{X-\mu}{\sigma}$

3.2) By 68-95-99.7 rule, 99.7% of the observations lies in the range between [-3,3]

Correct Answer : (c) [-3, 3]

3.3) We know that the definition of cumulative distribution function is

$F(x)=P[X\leq x]$

Therefore, it is a probability that X will take a value less than or equal to x.

Correct answer is : (b) less than or equal to x.

3.4) By central theorem, Let X1,X2...Xn are random sample from 'any distribution' then mean of sample follows normal distribution if sample size is large ( $n\rightarrow \infty$ ). Therefore,

Correct Answer : (c) Any