Delta Distribution

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What is a delta distribution?

The term delta distribution may refer to either the zero-modified lognormal distribution developed by Aitchison and Brown or the Dirac delta function.

Zero-modified lognormal distribution

The zero-modified lognormal (delta) distribution is a mixture distribution with a positive probability of a zero observation; positive observations (i.e., non-zero observations) have a log-normal distribution. The delta distribution is a special case of a zero-modified distribution.

The probability density function h(y;μ,σ,p) is:

PDF of the delta distribution.
PDF of the delta distribution.

Where f(x; μ,σ) is the density of a lognormal random variable X.

Note that the standard deviation (σ) and mean (μ) refer to the lognormal part of the mixture distribution.

Aitchinson and Brown [1] developed the “delta distribution” or zero-modified lognormal distribution to model economic data, although some researchers have applied the delta distribution to environmental data [2]. When sample data are non-negative and right-skewed, the minimum variance unbiased estimator (MVUE) of the delta distribution’s has been suggested as an alternative to the sample mean [3].

Dirac Delta Function / Distribution

The Dirac delta function isn’t a true “function” at all. The Dirac delta is an element of a set of mathematical objects called distributions – so the function is more aptly named a “delta distribution.”

The delta distribution describes real-world phenomena that can’t be described by regular functions. It  is widely used to describe analysis of concrete models (equations), signal transfer, signal boost, in picture detection and texture removal of pictures.  It is also used in a wide variety of natural sciences to describe white noise, the set of all sounds, or white light, the set of all light frequencies, or black holes, or delta shocks (from phenomenon such as earthquakes or tsunamis) [4].


[1] Aitchinson, J. and Brown, J. (1957). The lognormal distribution. Cambridge University Press, pages 87-99.

[2] Owen, W. & DeRouen, T. (1980). Estimation of the mean for lognormal data containing zeroes and left-censored values with applications to the measurement of worker exposure to air contaminants. Biometric, 36:707-719.

[3] Syrjala, S. Critique on the use of the delta distribution for the analysis of trawl survey data. ICES Journal of marine science. 57:831-842. 2000.

[4] Stevan Pilipovic, Djurdjica Takaci. On the delta distribution: Historical remarks, Visualization of delta sequences. Teaching mathematics and statistics in sciences. Retrieved June 27, 2022 from: