5.6b Mean-value and high-D labs

Source: Lab ideas from Computational Physics (Landau, Páez, Bordeianu), §5.6. Rewritten as original drills. Continues 5.6 Mean-value theorem and high-D Monte Carlo.

Practice the mean-value rule in 1D, then the 10D integral $I=\int_{[0,1]^{10}}(\sum x_i)^{2}\,d^{10}x=155/6$. Use the chapter LCG only.

Fixed LCG

r ← (1103515245 * r + 12345) % 2**31, uniform r / 2**31. Advance before each draw.

Formula refresh

Warm up: pick the matching equation among lookalikes (same walk as the theory lesson).

Labs

Activity: Mean-value 1D $e^{-x}$

Approximate $\int_0^1 e^{-x}\,dx$ with $N$ chapter-LCG samples (seed $1$). Print with 10 decimals.

Input: N.

Write-ups

0/4 solved

5.6b.W1

Using “mean”, “width”, and “average”, explain:

State the 1D mean-value Monte Carlo formula for ∫_a^b f.

Minimum 3 words. Your answer must include: “mean”, “width”, “average”.

0 words/ 3 min

Meet the writing requirements, then you can open the model answer.

Explorers

# mean_value_nd.py — 10D mean-value sketch (original)
M = 2**31

def ten_d(N, seed):
    r, acc = seed, 0.0
    for _ in range(N):
        s = 0.0
        for _k in range(10):
            r = (1103515245 * r + 12345) % M
            s += r / M
        acc += s * s
    return acc / N

Next: 5.7 Monte Carlo variance reduction.