Question

5. So far in our linear modeling, we have assumed that Ylz ~ NA,+Az,σ2); that is, there is a normal distribution of common variance around the regression line. Here, we change this up! Suppose that X~Unif(0, 1) and that for a given r, we know YlN(,22). (Here, the regression lne is 01z and the variance around the regression grows as r grows.) a. In R, figure out how to generate 1000 data points that follow this model and plot them. Draw the line yin red atop the points. Include your code and a sketch of the plot. (Hard!) b. Theoretically, find Cov(X, Y) and cor(X, Y). Check your theoretical answers by using your data from part a to find the sample correlation. Hint: Recall that the joint distribution of two random variables has a relationship to the condition and marginal distributions:

Answer 1

Solution:

8o o O og 2 0

> x=runif(1000,0,1)

> y=rnorm(1000,x,(x*x))

> lm(y~x)

Call:

lm(formula = y ~ x)

Coefficients:

(Intercept) x

0.01052 0.98437

> plot(y,x)

> abline(y,x)

> summary(lm(y~x))

Call:

lm(formula = y ~ x)

Residuals:

Min 1Q Median 3Q Max

-1.97628 -0.12499 -0.00986 0.12592 2.31425

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 0.01052 0.02840 0.37 0.711

x 0.98437 0.04935 19.95 <2e-16 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.442 on 998 degrees of freedom

Multiple R-squared: 0.285, Adjusted R-squared: 0.2843

F-statistic: 397.9 on 1 and 998 DF, p-value: < 2.2e-16