Data 604 - Week 6 Discussion

Problem 5

Given that the cdf is not defined under \(x \leq -3\) and \(x > 4\), we only need to concern ourselves with the range \(-3 < x \leq 4\). The generator function would transform to:

• \(R = \frac{1}{2} + \frac{x}{6} \rightarrow x = 6R-3\)
  • Under the range of \(-3 < 6R-3 \leq 0 \rightarrow 0 < R \leq \frac{1}{2}\)
• \(R = \frac{1}{2} + \frac{x^2}{32} \rightarrow x = \sqrt{32R - 16} \rightarrow 4\sqrt{2R-1}\)
  • Under the range of \(0 < 4\sqrt{2R-1} \leq 4 \rightarrow \frac{1}{2} < R \leq 1\)

\(\therefore\) our variable generator will have the form:

\[x =\begin{cases}6R - 3 & 0 < R \leq \frac{1}{2} \\\sqrt{32R-16} & \frac{1}{2} < R \leq 1\end{cases} \]

Using this model to generate 1000 samples produces the following histogram:

set.seed <- 8675309
R <- runif(1000)
x <- ifelse(R <= 1/2, 6*R-3, 4*sqrt(2*R-1))
nbin <- bigvis::find_width(x, 50)
nbin <- attr(nbin,"n")
ng <- data.frame(x,R)
ggplot(ng, aes(x)) + geom_histogram(bins = nbin) + xlab("x")