Problem 4 Verify the results of Problem 3 using R. Include your code and your printed output, and plots. iid Generate variables X1, X2, X3 Normal(0, 1) for a population of size N = 1000. Plot density...

Solve Problem 4

For 1) Just need code and explanation.

For 2) Just need code and explanation

Extracted text: Problem 4 Verify the results of Problem 3 using R. Include your code and your printed output, and plots. iid Generate variables X1, X2, X3 Normal(0, 1) for a population of size N = 1000. Plot density plots of each variable, compute the population mean and variance of each variable. Do these match what you would expect? 1) 2) For each construction of Z in Problem 3 ((a), (b), (c) and (d): • Create the variable Z as a function of X1, X2, X3 for the population of size N = 1000. Generate the variable from the distribution you found in Problem 3 for a population of size N = 1000. %3D • Plot the density plots for both variables. What do you observe? How do they compare? • Compute the mean and variance for both variables. What do you observe? • Based on the distribution you found in Problem 3, what should be the mean and variance of Z? Compare these to the numeric values you computed. Problem 3 iid Normal(0, 1). In each of the following cases, identify the distribution of the Let X1, X2, X3 random variable with the value of the associated parameter (e.g. don't only say Normal, you need to specify the mean and variance). a) Z = X1 +2X2 + 3X3. b) Z = x² + X3 + X3. c) Z = (X1 – X3)²/2. | d) Z = 2X3