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ID: past-exam-of-the-chemistry-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-1/7/b/solution

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Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 7 / b / Solution by

Codex  0 2026-10-20

A weighting function must first be normalized to a probability density. The constant choice on

[a, b]

means

q (x) = \frac{1}{b - a}, I = E_{q} [\frac{f ( X )}{q ( X )}] = (b - a) E_{q} [f (X)] .

(1)

Equivalently, transform

x = a + (b - a) z

and use the already normalized

g (z) = 1

[0, 1]

. Independent samples give the unbiased Monte Carlo integration estimator

I_{N} = \frac{b - a}{N} j = 1 \sum N f (a + (b - a) U_{j}), U_{j} \sim Uniform (0, 1) .

(2)

function uniform_monte_carlo(f, a, b, N):
    require N >= 2
    total = 0
    total_squared = 0
    for j in 1,...,N:
        u = independent_uniform_0_1()
        value = f(a+(b-a)*u)
        total += value
        total_squared += value*value
    average = total/N
    sample_variance = (total_squared-N*average*average)/(N-1)
    integral = (b-a)*average
    standard_error = abs(b-a)*sqrt(sample_variance/N)
    return integral, standard_error

For a general positive normalized importance density

q

, replace each sample contribution by

f (X_{j}) / q (X_{j})

with

X_{j} \sim q

. Here the uniform choice is the special case of importance sampling that assigns equal weight to every region of equal length.

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