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2. Consider estimating = =√₂(x - 2)e-dx.

(a) It is known that 0 = E[ƒ(X)], where X ~ Exp(1). What is f(.)?

(b) Write the pseudo-codes of an algorithm that generates variates from Exp(1) con-

ditional on being valued in the interval [2, ∞).

(c) Write the pseudo-codes of an algorithm that estimates with antithetic variate,

where there are 1000 variates in total.

(d) Write the pseudo-codes of an algorithm that estimates with stratified sam-

pling, where 5 strata are considered and 200 data are sampled from each stratum.

(e) Write the pseudo-codes of an algorithm that estimates with control variate,

where we use the variates in part (b) as control variates (again, 1000 variates).

Fig: 1