Past exam of the simulation and statistical programming course of the University of Oxford / 2021 / 1 / b

Solution

E [e^{t X_{j}}] = e^{t^{2} /2}

. Using independence of the coordinates,

θ = E_{p} [ϕ (X)] = \prod_{j = 1}^{d} E [e^{- X_{j}}] = e^{d /2}, E_{p} [ϕ (X)^{2}] = \prod_{j = 1}^{d} E [e^{- 2 X_{j}}] = e^{2 d} .

Var (θ_{n}^{RS}) = \frac{e ^{2 d} - e ^{d}}{n} .

This counts

n

accepted draws, requiring

n σ^{d}

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