Question: as Suppose X is a normal variable with mean and variance 2. Let f(x) be the density function of X. For a constant 0,

as Suppose X is a normal variable with mean and variance 2.

as Suppose X is a normal variable with mean and variance 2. Let f(x) be the density function of X. For a constant 0, define the tilted density function fo(x) = f(x)ex Mo where Me is the normalization constant to ensure that fo() is a probability density function. (a) Find the expression of M, in terms of 0, and . (b) What kind is the tilted distribution corresponding to fo(x)? (c) Suppose =0 and = 1. Our goal is to estimate P(X > 10) which is a rare event prob- ability. Design a importance sampling algorithm using the tilted distribution. Specify your choice of 0 and explain why it leads to variance reduction.

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