Homework Problem Set 8, Due Monday October 9, 2017.
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Assigned       9/29/2017,  due 10/9:       14 points

(a) Give R code to generate (and then test the distribution in order 
to check your work) 5,000 pseudo-random variates from the density
on [0,7] proportional to:

1/(1+0.1*x^2)^2 + x^3 *exp(-x) + 0.2*(abs(x-3)<2) , 0 < x < 7


(b) Show by simulation of 10000 independent data batches of size 50
that the maximum likelihood estimators (MLEs) of each of the two 
parameters  a,b   from a  dlogis(x,a,b) density (based on a sample of 
size 50) is approximately normal, with the mean and variance predicted 
from the Fisher Information for this estimation problem.

Can you detect any evidence of non-normality in the distributions of 
these MLEs?


(c) Generate 10000 independent random variables with density 
proportional to   (1+2*x)*x^2*(1-x)^3 on (0,1). Do this by an 
efficient Accept-Reject method.