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TU München Zentrum Mathematik Lehrstuhl M16 E. Sonnendrücker K. Kormann Sommersemester 2014 Monte Carlo Methods with applications to plasma physics Exercise sheet 7 1. Importance sampling 2 Let us consider the PDF f (x) = √12π e−x /2 of the standard normal distribution. We want to √ compute E[ 2πχ[0,1] (X)] using Monte–Carlo integration. √ (a) Estimate E[ 2πχ[0,1] (X)] by Monte–Carlo integration with random numbers X drawn from the standard normal distribution. Compute the mean squared error. (b) As an alternative consider sampling U from the uniform distribution on [0, 1] with PDF 2 χ[0,1] (x). In this case, we want to compute E[e−U /2 ]. Compute the mean squared error and compare to the result in (a). 2. Control variates Let us consider the random variable X := U 2 + εU with U uniformly distributed on [0, 1]. Now assume that we know the expected value of Y := U 2 so that we can use it as a control variate. As a function of ε compute the variance of X and Z := X − (Y − E[Y ]).