Question 23.5: Suppose that the behavior of a nuclear particle is governed ......
Suppose that the behavior of a nuclear particle is governed by a normal probability distribution in which the mean (or average) value of the energy in the distribution is 0.5 MeV and the energy of the particle in question is 1.0 MeV. Calculate the standard deviation and the variance of the energy of the particle from the mean. If the distribution contains 1 million particles, and the variance of the entire distribution is 0.25 MeV, estimate the value of the fractional error E that a Monte Carlo calculation would have if it were applied to this distribution.
The standard deviation of the particle is given by σ = |v − ⟨v⟩| = |1.0 − 0.50| = 0.50 MeV. The variance V of the energy of the particle is the standard deviation squared or V = σ² = |v − ⟨v⟩|² = 0.25 MeV. The fractional error E in a Monte Carlo calculation that uses this distribution is given by E = \sqrt{[(1/N).V/(v)^2]}. In this case, N = 1,000,000, V = 0.25 MeV, and ⟨v⟩ = 0.50 MeV, so E = 0.001 = 0.1%.