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- Question : 1TQ - Suppose that n pairs of outcomes (xi, yi) of the variables x and y have been observed. a. Prove that the sample correlation coefficient between the variables x and y always lies between 1 and
- Question : 2TQ - a. Prove the results in (1.15) and (1.16). b. Suppose that x and y are independent random variables. Prove that the conditional distribution of yjx
- Question : 3TQ - a. Prove the result in (1.10) for the case of a linear transformation z
- Question : 4TQ - a. Show that the mean and variance of the Bernoulli distribution are equal to p and p(1 p) respectively. b. Show that the mean and variance of the binomial distribution are equal to np and np(1 p) respectively. c. Show the result in (1.23) for the case that A is an n n non-singular matrix, by using the generalization of the result in (1.19) to the case of n functions. d. Show that, when two jointly normally distributed random variables are uncorrelated, they are also independent. e. Show that the first four moments of the normal distribution N(m, s2) are equal to m1
- Question : 5TQ - Let y N(0, I) be an n 1 vector of independent standard normal random variables and let z0
- Question : 6TQ - Let yi NID(m, s2), i
- Question : 7TQ - a. Prove the equality in (1.45) for arbitrary distributions. b. Prove the inequality of Chebyshev, which states that for a random variable y with mean m and variance s2 there holds P[jy mj cs] 1=c2 for every c > 0. c. Use the inequality of Chebyshev to prove that the two conditions in (1.48) imply consistency. d. Use this result to prove that the maximum likelihood estimators ^mML and ^s2 ML in Example 1.8 are consistent. e. Prove the four rules for probability limits that are stated in the text below formula (1.48)
- Question : 8TQ - Let y1, , yn be a random sample from a Bernoulli distribution. a. Derive the maximum likelihood estimator of the parameter p
- Question : 9TQ - Let yi IID(m, s2), i
- Question : 10TQ - Let y1, , yn be a random sample from a population with density function fy(v)
- Question : 11SQ - In this exercise we consider data of ten randomly drawn students (the observation index i indicates the position of the students in the file of all 609 students of Example 1.1). The values of FGPA, SATM, and FEM of these students are as follows. a. Compute for each of the three variables the sample mean, median, standard deviation, skewness, and kurtosis. b. Compute the sample covariances and sample correlation coefficients between these three variables. c. Make three histograms and three scatter plots for these three variables. d. Relate the outcomes in a and b with the results in c. e. Compute the conditional means of FGPA and SATM for the four male students and also for the six female students. Check the relation (1.15) (applied to the
- Question : 12SQ - Consider the data set of ten observations used in Exercise 1.11. The FGPA scores are assumed to be independently normally distributed with mean m and variance s2, and the gender variable FEM is assumed to be independently Bernoulli distributed with parameter p
- Question : 13SQ - In this exercise we consider data of 474 employees (working in the banking sector) on the variables y (the yearly salary in dollars) and x (the number of finished years of education). a. Make histograms of the variables x and y and make a scatter plot of y against x. b. Compute mean, median, and standard deviation of the variables x and y and compute the correlation between x and y. Check that the distribution of the salaries y is very skewed and has excess kurtosis. c. Compute a 95% interval estimate of the mean of the variable y, assuming that the salaries are NID(m, s2). d. Define the random variable z
- Question : 14SQ - In this simulation exercise we consider the quality of the asymptotic interval estimates discussed in Section 1.4.3. As data generating process we consider the t(3) distribution that has mean equal to zero and variance equal to three. We focus on the construction of interval estimates of the mean and on corresponding tests. a. Generate a sample of n
- Question : 15SQ - In this simulation exercise we consider an example of the use of the bootstrap in constructing an interval estimate of the median. If the median is taken as a measure of location of a distribution f , this can be estimated by the sample median. For a random sample of size n, the sample median has a standard deviation of (2f (m) ffiffiffi pn)1 where m is the median of the density f. When the distribution f is unknown, this expression cannot be used to construct an interval estimate. a. Show that for random samples from the normal distribution the standard deviation of the sample median is s ffiffiffi pp= ffiffiffiffiffiffi p2n whereas the standard deviation of the sample mean is s= ffiffiffi pn. Comment on these results. b. Show that for random samples from the Cauchy distribution (that is, the t(1) distribution) the standard deviation of the sample mean does not exist, but that the standard deviation of the sample median is finite and equal to p=(2 ffiffiffinp ). c. Simulate a data set of n

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