Math 304: Homework 6
1. Approximate log2 by using a uniform (0,1) generator. Obtain an error estimation in terms
of a large sample 95% CI. Using R, obtain your estimate for 10,000 simulation and compare
it to the true value.
Hint: Recall that log 2 = R1
0
1
x+1 dx.
2. Similar to the last exercise but now approximate
Z1.96
0
1
√2πexp{−t2/2}dt.
3. Determine a method to generate random observations for the following pdf
f(x) = 4x3.
Write an R function to generate a random sample of size nobservations from this pdf. Check
that your function works by calculating a confidence interval for the mean of the distribution.
4. Write an R function to generate random of nobservations for the logistic pdf:
f(x) = e−x
(1 + e−x)2,−∞ < x < ∞.
Obtain a sample from this distribution and graph the histogram. Overlay a plot of the pdf.
5. Write an R function that uses Gibbs algorithm to generate standard normal random variables.
If
f(x, y) = 1
√2π,−∞ < x < ∞,0< y < e−x2/2
the marginal density of Xis standard normal. The conditional distributions are Y|X=x∼
U(0, ex2/2), and X|Y=y∼U(−√−2 log y, √−2 log y). Start with a number in the support
of X. Obtain a sample from this distribution and graph the histogram. Overlay a plot of the
pdf.
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