Repository navigation
[BUG]: stats/base/dists/geometric/mgf returns the MGF for the number of trials instead of the number of failures #15780
Description
Description
stdlib's geometric distribution counts the number of failures before the first success (support 0, 1, 2, ...), per the pmf, mean, and variance packages. Its MGF should be p / (1 - (1-p) e^t), but geometric/mgf returns p e^t / (1 - (1-p) e^t), which is the MGF for the number of trials. Every value is off by a factor of e^t.
Reproduction
var mgf = require( '@stdlib/stats/base/dists/geometric/mgf' );
var pmf = require( '@stdlib/stats/base/dists/geometric/pmf' );
var mean = require( '@stdlib/stats/base/dists/geometric/mean' );
var s = 0.0;
var k;
for ( k = 0; k < 2000; k++ ) {
s += Math.exp( 0.1*k ) * pmf( k, 0.3 );
}
console.log( mgf( 0.1, 0.3 ), s );
// => 1.4646 1.3252
console.log( ( mgf( 1e-6, 0.3 ) - mgf( -1e-6, 0.3 ) ) / 2e-6, mean( 0.3 ) );
// => 3.3333 2.3333
console.log( mgf( 0.5, 1.0 ) );
// => 1.6487 (X is always 0 when p = 1, so this should be 1)Expected Results
mgf( 0.1, 0.3 ) ≈ 1.3252, and mgf( t, 1.0 ) = 1 for all valid t. The negative binomial MGF in stdlib, (p / (1 - (1-p) e^t))^r, already uses the failures convention. With r = 1 it gives the correct geometric value.
Actual Results
Values are too large by a factor of e^t. The Julia fixtures were generated with Distributions.jl 0.23.8. That version had the same trials-based formula. Current Distributions.jl evaluates p / (p - (1-p) expm1(t)).
Version
develop (25751e9)
Environments
Node.js
Activity
stdlib-bot commentedon Oct 2, 2026
👋 Hi there! 👋
And thank you for opening your first issue! We will get back to you shortly. 🏃 💨
If you have any further questions while waiting for a response, please join our Zulip community to chat with project maintainers and other community members.
kgryte commentedon Oct 2, 2026
@SajalDevX I am not sure you are correct here. There are two formulations of the geometric distribution (see Wikipedia: https://en.wikipedia.org/wiki/Geometric_distribution). Both are correct.
@Planeshifter Would you be willing to comment here?
SajalDevX commentedon Oct 2, 2026
Thanks for taking a look! Agreed that both parameterizations are valid. The reason I flagged this is that the rest of the
geometricnamespace uses the number-of-failures form:pmfis(1-p)^x pforx = 0, 1, 2, ...,meanis(1-p)/p, andcdfis1-(1-p)^(floor(x)+1).mgfis the only one using the number-of-trials form, so it doesn't match the other functions in the package.For example, with
p = 0.3, the derivative of the currentmgfatt = 0is1/p ≈ 3.33, whilemean( 0.3 )returns0.7/0.3 ≈ 2.33. Summingexp(t*x) * pmf(x, p)directly fort = 0.1, p = 0.3gives1.3252, which matchesp / (1 - (1-p)e^t), while the current implementation returns about1.4646.If you'd rather keep the current formula, I'm happy to change the PR to only update the README/docs to say it describes the number of trials. But that would leave
mgfdescribing a different random variable than the rest of the package.kgryte commentedon Oct 2, 2026
I'll defer to @Planeshifter here.
Planeshifter commentedon Oct 2, 2026
Yes, this report is accurate. Julia's Distribution package had the same parameterization inconsistency, causing us to carry it forward. They fixed it in https://github.com/JuliaStats/Distributions.jl/pull/1604. Per the linked PR, it would be worthwhile to also confirm whether the same issue is present in the negative binomial MGF package.