BINOMDIST: Binomial Distribution CDF

Binomial distribution CDF counts the probability of success in fixed trials — discrete probability at its most fundamental.

Property	Value
Category	Numeric
Inputs	Source (close)
Parameters	`period` (default 50), `trials` (default 20), `threshold` (default 10)
Outputs	Single series (Binomdist)
Output range	Varies (see docs)
Warmup	`period` bars
PineScript	binomdist.pine

BINOMDIST computes the cumulative distribution function of the Binomial distribution, mapping a min-max normalized price to a success probability $...
Trading note: Binomial distribution; models success/failure sequences. Win streak probability in trading.
Validated against TA-Lib, Skender, and Tulip reference implementations where available.

BINOMDIST computes the cumulative distribution function of the Binomial distribution, mapping a min-max normalized price to a success probability p and evaluating P(X \leq k) for X \sim \text{Binomial}(n, p). The normalized price position within its lookback range determines the probability of success per trial, while the trial count n and threshold k control the shape of the CDF response. The output is a [0, 1] bounded oscillator where values near 0 indicate the price-derived probability makes k or fewer successes very unlikely (bullish pressure), and values near 1 indicate k successes are very likely (established range).

Historical Context

The Binomial distribution, formalized by Jakob Bernoulli in 1713 and refined by Abraham de Moivre, is the foundational discrete probability distribution for counting successes in independent trials. Its CDF application to financial time series transforms the continuous price position into a discrete probabilistic framework: "given the current price's relative position as a probability, how likely is it that at most k out of n events would succeed?" This reframing provides a nonlinear transformation that is particularly sensitive around the probability values where k/n transitions from unlikely to likely. The log-space summation technique used here avoids factorial overflow for large n, leveraging the Lanczos log-gamma approximation for \ln(n!) computation.

Architecture & Physics

Two-Stage Pipeline

Min-Max Normalization: The source is normalized to p \in [0, 1] over the lookback window. This probability represents the "success rate" implied by the price's position within its recent range.
Binomial CDF Summation: The CDF P(X \leq k) is computed as a direct sum of binomial probabilities from i = 0 to k. Each term is computed in log-space to avoid overflow: \ln\binom{n}{i} + i\ln(p) + (n-i)\ln(1-p), then exponentiated and accumulated. The log-binomial coefficient uses the Lanczos log-gamma function.

Edge Cases

p \leq 0: All mass at X = 0, so P(X \leq k) = 1 for any k \geq 0
p \geq 1: All mass at X = n, so P(X \leq k) = 1 only if k \geq n
Result is clamped to [0, 1] to guard against floating-point accumulation drift

Mathematical Foundation

Binomial CDF:

P(X \leq k) = \sum_{i=0}^{k} \binom{n}{i} p^i (1-p)^{n-i}

Log-space computation (avoids factorial overflow):

\ln\binom{n}{i} = \ln\Gamma(n+1) - \ln\Gamma(i+1) - \ln\Gamma(n-i+1) P(X \leq k) = \sum_{i=0}^{k} \exp\!\left[\ln\binom{n}{i} + i\ln(p) + (n-i)\ln(1-p)\right]

Lanczos log-gamma (g = 7, 9 coefficients): same as BETADIST.

Default parameters: period = 50, trials = 20, threshold = 10 (symmetric: k = n/2).

Performance Profile

Operation Count (Streaming Mode)

Binomial distribution PMF/CDF uses log-gamma for large n; direct factorial for small n.

Operation	Count	Cost (cycles)	Subtotal
Input validation (n, k integers; p in [0,1])	3	2 cy	~6 cy
Log-binomial coefficient via log-Gamma	2	25 cy	~50 cy
k * log(p) + (n-k) * log(1-p)	2	8 cy	~16 cy
exp() for PMF	1	20 cy	~20 cy
CDF sum over k terms (optional)	k	90 cy	~90k cy
Total (PMF only)	O(1)	—	~92 cy

PMF is O(1); CDF requires summing k+1 PMF values — O(k) where k = successes. For large cumulative queries, use regularized incomplete beta instead.

Batch Mode (SIMD Analysis)

Operation	Vectorizable?	Notes
Log-Gamma computation	No	Transcendental; scalar
exp() for PMF	Partial	_mm256_exp_pd with SVML
CDF accumulation	No	Sequential sum dependency

PMF batch can use SVML exp vectorization. CDF must remain scalar.

Resources

Bernoulli, J. (1713). Ars Conjectandi
Press, W. et al. (2007). Numerical Recipes, 3rd ed., §6.2 (Incomplete Beta as alternative)
PineScript reference: binomdist.pine

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