By B. Gnedenko
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1. SAMPLING AND MONTE CARLO INTEGRATION 49 with Gaussian densities with zero mean and variance σ 2 . Let η1 = −2σ 2 log ξ1 cos(2πξ2), η2 = −2σ 2 log ξ1 sin(2πξ2), where ξ1 and ξ2 are equidistributed in [0, 1]; then η1 , η2 are Gaussian variables with means zero and variances σ 2 , as one can see from the identity ∂η1 ∂η1 −1 ∂ξ1 ∂ξ2 | ∂η |dη1 dη2 = dξ1 dξ2 ∂η2 2 ∂ξ1 ∂ξ2 (the short outer vertical lines denote an absolute value, while the tall inner vertical lines denote a determinant), which becomes, with the equations above, η12 + η22 1 exp − 2πσ 2 2σ 2 dη1 dη2 = dξ1 dξ2.
Suppose η is a random variable on Ω. Then the average of η given A is E[η|A] = Thus if η = η(ω)P (dω|A). ci χBi , then E[η|A] = ci χBi (ω)P (dω|A) = ci P (Bi |A). 3. CONDITIONAL PROBABILITY AND CONDITIONAL EXPECTATION 37 Example. Suppose we throw a die. Let η be the value of the top face of the die. 5. i=1 Suppose we know that the outcome is odd. Then the probability that the outcome is 1, given this information, is P ({1}|outcome is odd] = P ({1} ∩ {1, 3, 5}) 1/6 1 = = ; P ({1, 3, 5}) 1/2 3 and the average of η given A = {1, 3, 5} is 1 E[η|outcome is odd] = (1 + 3 + 5) = 3.
4] A. Kolmogorov and S. Fomin, Elements of the Theory of Functions and Real Analysis, Dover, New York, 2000. [5] P. Lax, Linear Algebra, Wiley, New York, 1997. 1. ” What does this mean? ” To make sense of this, we formalize the notions of experimental outcome, event, and probability. Suppose that you make an experiment and imagine all possible outcomes. Definition. A sample space Ω is the space of all possible outcomes of an experiment. For example, if the experiment is “waiting until tomorrow, and then observing the weather,” Ω is the set of all possible weathers tomorrow.
The Theory of Probability by B. Gnedenko
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