Question

PROBLEM #1:TRUE of FALSE

For each of the following statements, determine whether it is true or false. Label “T” if it is true, otherwise label “F”.

1. 1-a. If two distributions have the same moment-generating function, then they are identical at almost all points. For a random variable X, if its moments of order k (k > 0) exist, then its moment generating function is continuously differentiable up to order k.

2. 1-b. LetXbearandomvariablewiththeprobabilitydensityfunctionp(x)=ca/(a+x2) where c, a are positive constants and a > 2. The moment generating function of X does not exist.

3. 1-c. Point estimators constructed via maximum likelihood estimation are MVUE.

4. 1-d. Point estimators are statistics which correspond to parameters in the population. For the population mean μ, its point estimator is the sample mean X and for the population proportion p, its point estimator is the sample proportion pˆ.

5. 1-e. For a constructed 95% confidence interval for the mean, it can be interpreted as follows: if we take large random samples over and over again from the same population, then 95% of the resulting intervals will cover the sample mean.

6. 1-f. Assuming that we are given a random sample X1, · · · , Xn and we are interested in constructing a 90% confidence interval for the population mean, it is necessary that the distribution of the variable of interest follows a normal curve or a t-curve.

7. 1-g. One can interpret the width of a confidence interval as the precision of the interval estimator; one can interpret the associated confidence level as the reliability of the interval estimator. The width is a function of the sample size and when the sample size goes larger, the width becomes smaller.

8. 1-h. When performing regression tasks using the simple linear regression model, the variability of the response variable comes from that of the noise variable and the variance of the two variables are exactly the same.

9. 1-i. In simple linear regression, the distribution of the independent variable X as well as the dependent variable Y are assumed to be normally distributed. This also applies to the noise variable which is usually assumed to be drawn from a normal distribution with zero mean and variance σ2 for each fixed X = x.

10. 1-j. In a hypothesis test, Type I error is the error made when the null hypothesis is rejected when in fact the null hypothesis is true; Type II error is the error made when the null hypothesis is not rejected when it is false. The probability of making the two types of errors are inversely related.

Answer 1

1-a. T (Both the statements are true. These are the properties of the MGF)

1-b. T. The reason is as follows:

1-c. F. A counter example is the MLE of the variance of a normal population based on a SRSWR sample. It is the sample variance 'n' in the denominator and is not unbiased, let alone being MVUE.

1-d. F. Statistic don't need to correspond to the population parameter by definition. Any function of sample that doesn't depend on the parameter in it's form is a statistic. Thus , is also a statistic for estimating parameter . It is a 'bad' statistic, but a statistic.

1-e. T. This follows from SLLN. Let p be the probability that the interval contains the mean. Then p = 0.95 (since it's a 95% CI). Thus, due to SLLN, if we evaluate the intervals a large number of times. Approx 95% of them would contain the mean

1-f. F. We can construct CI even if the distribution is anything different.

1-g. T. The statements stated are true. Only the last statement might not be true always. But most of the times, with increasing sample size, the width of CI decreases.

1-h. T. These are the basic assumptions of simple linear regression. We assume that the variability is due to the random noise which is homoscedastic.

1-i F. We do not assume any distribution for the independent variable. We assume it is given and do our analysis considering this.

1-j. T. The statements are true. Only the last statement is sometimes not true when the support of the distribution under null and alternative are disjoint. In that case they are not inversely related and it's possible to minimise both of them. However, such situations are extremely trivial and are only theoretical possibility. They don't occur in real life. For all practical purposes, the statement is true.