Assignment_4_Solution - Statistics 371 Spring 2007...

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1 Statistics 371 Spring 2007 Assignment 4 Solutions 1. We can think of simulations, (often called Monte Carlo Methods) as applications of SRS. These methods are often used to evaluate an integral in high dimensions when the integrand is expensive to compute. The integral correspond to expected values of quantities of interest in the simulation. Here is a toy example estimating a definite integral. Suppose we want to evaluate Af t d t t dt == zz () exp( . ) 0 1 2 0 1 05 2 π . This is the probability that a standard gaussian random variable falls in the interval (,) 01 so we know that A=0.3413 . Consider the very large frame Uu u N = ( ,. ..., ) 1 that is the finite set of possible values returned by a uniform 01 random number generator and let y f u ii = ( ). Then the population average y N i U is very close to the integral. Any difference is frame error. a) Use the code u<-runif(100) to generate a simple random sample of 100 units from the frame. Calculate f<-exp(-.5*u*u)/sqrt(2*pi). Hence find an estimate of A and the corresponding 95% confidence interval. The estimate is the sample average and the corresponding 95% confidence interval is ˆ 1.96 1000 average σ ± where ˆ is the sample standard deviation. Note that the factor correcting for the finite population is immaterial here. My CI is 0.3421 0.0030 ± which contains the (known) value. b) What use is the confidence interval in this context? The confidence interval gives us a very cheap way to assess the approximation. c) How large a sample is required to estimate A so that the 99% confidence interval has length 0.002?
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This note was uploaded on 05/12/2011 for the course STAT 371 taught by Professor Ahmed during the Fall '09 term at Waterloo.

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Assignment_4_Solution - Statistics 371 Spring 2007...

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