Homework 7

Problem 1

Write a routine that generates random variates from a distribution with density proportional to $$g(x)=\sqrt{(1+x)^3(1-x)^5}$$ $-1<x<1$. Use only runif in your routine. Show that your routine generates the correct data.

Problem 2

Write a routine that generates random variates from a distribution with density proportional to $$g(k)=\exp(-\sqrt{k})$$ $k=0, 1, ...$. Show that your routine generates the correct data.

Problem 3

Write a routine that generates random variates from a distribution with density $$f(x,y)=1.732\exp\left(-[2+\sin(2\pi x)] y\right)$$ $0<x<1$, $y>0$. Use the simulated data to find F_X(0.5), F(0.5, 1), E[X], Var[X], E[Y], Var[Y], Cor(X,Y), F_X|Y=1(0.5|1) and E[X|Y=1].

BONUS: Draw the graph of f_X|Y=0.1(x|0.1).