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Probability, Entropy, and MC Simulation
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Monte Carlo simulation error typically decreases with the number of runs N as:
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Entropy H(X) for a discrete variable with n outcomes is maximized when:
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Given the joint pmf: P(X=0,Y=0)=0.2, P(X=0,Y=1)=0.3, P(X=1,Y=0)=0.1, P(X=1,Y=1)=0.4. Find P(X=0).
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The core principle of Monte Carlo simulation is to:
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If X_i are i.i.d. with mean μ=10 and variance σ²=4, the approximate distribution of the sample mean for n=100 according to CLT is:
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A key property of the Multivariate Normal Distribution is:
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If Cov(X, Y) = 0, then X and Y are:
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The "Law of Total Expectation" states that:
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For a discrete random variable, the probability mass function (pmf) must satisfy:
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The moment generating function (MGF) is defined as:
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