Download Monte Carlo Methods with applications to plasma physics

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 ]).