Homework 8, Due Wednesday October 7.
-----------------------------------

Consider the problem of calculating the exact probability 
for the sum of 51  i.i.d. observations from a density

f(x) =  0.5 * exp(x) * I[x<0] + exp(-x^2/2) * I[x>0]/sqrt(2*pi)

to be greater than 15 .

(a) Find this probability with confidence-interval half-width 
less than .0002) with confidence level at least 99%.

## Use an importance-sampling approach, using as candidate
##   joint density for each X of the form g(x) = c exp(lambda*x) f(x)
##   where c is defined so that g integrates to 1, and lambda is
##   chosen by any method you like.

(b) Find the best lambda, i.e., the one which will give the narrowest
confidence intervals, either by trial and error [NOT the 
preferred way] or by setting up an optimization problem based 
on numerical integration.] Such an optimization can be performed
numerically using the R function  "optim" .