Light arrives as countable photons, and counting is noisy: a dim scene is grainy not because the sensor is bad but because few photons landed. Gather more light and the signal-to-noise climbs as the square root — the machine’s sight has a floor set by physics. Slide the exposure (or tap) and watch the grain settle.
Photon shot noise, the fundamental limit of any imager. The number of photons hitting a pixel in a fixed time is Poisson-distributed with mean λ; a defining property of the Poisson distribution is that its variance equals its mean, so the noise standard deviation is √λ and the signal-to-noise ratio is λ/√λ = √λ. Quadrupling the exposure only doubles the SNR — the diminishing return that governs every low-light photograph and every faint-star exposure. Real Poisson samples (Knuth's algorithm, seeded) are drawn across a brightness ramp; the sample variance is checked against the mean live. A fail-loud self-check throws unless the variance tracks the mean and the SNR grows as √λ.
Pure shot noise on a fixed brightness ramp (real sensors add read noise, dark current and a gain); λ is a demonstration scale. The Poisson sampling, the variance=mean identity and √λ are computed exactly.