4.3 Testing random distributions

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §4.4. Rewritten in original wording. Continues 4.2b Decay labs. Labs: 4.3b Random-distribution test labs.

A computer’s “random” stream follows a definite rule, so successive values are correlated. That can poison a Monte Carlo that assumes independent draws. Habit: run a few uniformity / randomness checks — ideally alongside the simulation that depends on them.

Look first

The cheapest test is to look. A uniform generator on $[0,1)$ should spit values that clearly sit in that interval, differ from each other, and show no cartoon pattern like $0.3333$ forever.

Sample from the chapter LCG (r ← (1103515245 r + 12345) % 2³¹, then $r/2^{31}$), seed $1$:

0.513870	0.175741	0.308652	0.534534
0.947628	0.171736	0.702231	0.226431
0.494773	0.124720	0.083899	0.389647

Your eye already sees $[0,1)$ coverage and no short repeating motif. That is not a proof — only a smoke test.

Visual structure tests

1. $r_i$ versus $i$ (Fig. 4.2 style). Expect a cloud filling the strip $0\le r<1$ with jitter, not a smooth curve.
2. Successive-pair scatter $(r_i,r_{i+1})$ or $(r_{2i},r_{2i+1})$ (Fig. 4.1 style). Random uniforms fill the unit square as a featureless cloud. A lattice or stripes ⇒ reject for serious work.

Moment test for uniformity

Walk the sample-moment definition, the exact $\int_0^1 x^k=1/(k+1)$, the lag product $C(k)$, and its $1/4$ prediction — pick the matching equation at each step.

Practical checklist: $k=1,3,7$ and $N=10^2,10^4,10^5$; confirm $\sqrt{N}|\langle x^k\rangle-1/(k+1)|$ stays order one.

Lag-product correlation $C(k)$

Near-neighbor dependence shows up in the lag product from the walk above. Matching $1/4$ with $\sqrt{N}|C-1/4|=O(1)$ supports independence at lag $k$. Still plot the scatter: a bad LCG can pass crude averages while failing the eye test.

Quiz

RNG tests

1 / 4

For uniform[0,1] draws, ⟨x³⟩ should approach…

Pick an answer to see if you are right.

Next: 4.3b Random-distribution test labs — moment and $C(k)$ drills on the chapter LCG.