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ass4sol - Assignment 4 Solution 1 We can think of...

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Assignment 4 Solution 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 the definite integral 1 0 ( ) A f u du = where 1/2 ( ) exp( ) f u u u = . a) Explain why where U is a uniform random variable on the interval (0,1) 1/2 [ exp{ }] A E U U = This is the definition of expected value when the probability density function of U is . ( ) 1 0 1 f u u = < < b) Consider the very large frame U u u N = ( , .... , ) 1 that is the finite set of possible values returned by a uniform ( , random number generator and let . Explain why the frame average is close to the integral. We consider any difference as frame error. ) 0 1 1/2 exp{ } i i i y u u = 1 / N i i y N μ = = If we divide the interval (0,1) into N sub-intervals of length 1/ when N is large, then from the definition of the integral N μ will be close to its value. c) Use the R code y<-runif(100) to generate a simple random sample of 100 units from the frame. Find an estimate of μ and the corresponding 95% confidence interval. My set of 100 uniform (0,1) values produced the estimate ˆ 0.3810 μ = with standard error .0075. The 95% confidence interval is 0.3810 0.0146 ± {your answer will be different} d) What use is the confidence interval in this context? The confidence interval provides an estimate of the possible error in the approximation.
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