Simulate two Normally distributed random variables X1 and X2 with correlation 0.8, both should have a mean value of 70 and a standard deviation of 8. equation 5.3.2 for formulas that will help you see how to do the simulation. There are other ways to do this as well. Think of X2 as a score on the second test in a class, and X1 as the score on the first exam.a. Analytically find the expected value and variance of `Change = X2 - X1`.b. Use a simulation with n=1000 to find the mean and variance of `Change` using simulation.c. Plot your simulated data on a scatterplot.d. Also plot X1 vs. Change.e. How might regression to the mean cause issues in assessing whether or not a student improved or did worse on the second test compared to the first? ## 5.3.2 Bivariate Normal DistributionThe normal distribution plays a critical role in probability theory and is widely used across various applications. We've previously discussed a single normal random variable, and now we expand to two or more normal random variables. As seen in Theorem 5.2, the sum of two independent normal random variables is also normal. However, when the two normal random variables are dependent, their sum may not be normal, as illustrated in the following counterexample:### Example 5.35Let \(X \sim N(0,1)\) and \(W \sim \text{Bernoulli}\left( \frac{1}{2} \right)\) be independent random variables. Define the random variable \(Y\) as a function of \(X\) and \(W\):\[Y = h(X,W) = \begin{cases} X & \text{if } W = 0 \\ -X & \text{if } W = 1 \end{cases}\]Since \(N(0,1)\) is symmetric around zero, \(-X\) is also \(N(0,1)\). We can express:\[F_Y(y) = P(Y \leq y) = P(Y \leq y | W = 0)P(W = 0) + P(Y \leq y | W = 1)P(W = 1)\]This results in:\[= \frac{1}{2} P(X \leq y) + \frac{1}{2} P(-X \leq y)\]As \(X\) and \(-X\) are \(N(0,1)\):\[= \frac{1}{2}\Phi(y) + \frac{1}{2}\Phi(y) = \Phi(y)\]Hence, \(Y \sim N(0,1)\). Now consider:\[Z = X + Y = \begin{cases}2X & \text{with probability } \frac{1}{2} \\0 & \text{with probability } \frac{1}{2}\end{cases}\]Thus, \(Z\) is a mixed random variable with its PDF given by:\[f_Z(z) = \frac{1}{2} \delta(z) +

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MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Simulate two Normally distributed random variables X1 and X2 with correlation 0.8, both should have a mean value of 70 and a standard deviation of 8. equation 5.3.2 for formulas that will help you see how to do the simulation. There are other ways to do this as well. Think of X2 as a score on the second test in a class, and X1 as the score on the first exam.

a. Analytically find the expected value and variance of `Change = X2 - X1`.

b. Use a simulation with n=1000 to find the mean and variance of `Change` using simulation.

c. Plot your simulated data on a scatterplot.

d. Also plot X1 vs. Change.

e. How might regression to the mean cause issues in assessing whether or not a student improved or did worse on the second test compared to the first?