# Nakagami Distribution (Nakagami-m): Definition

Probability Distributions > Nakagami Distribution

## What is the Nakagami Distribution?

The Nakagami distribution (also called the Nakagami-m or Nakagami-μ distribution) is a fairly new probability distribution, first appearing in 1960. It is a generalized way to model small scale fading for dense signal scatters and it one of the most common distributions for modeling right-skewed, positive data sets. Real world applications include modeling wireless signal and radio wave propagation, characterizing breast tumors using ultrasound imaging and in meteorology.

The probability density function formula is:

Where:

The two parameters combined determine the height, steepness and concavity of the probability density curve. The distribution is only valid on the interval (0,∞). In other words, if x ≤ 0, then the probability is zero.

Family of Nakagami Distributions showing the effects of different shape and scale parameters. Image: Krishnaveda | Wikimedia Commons.

The tails are usually thin, meaning that the tail decreases exponentially for large x-values.

## Relation to Other Distributions

The distribution is related the the gamma distribution: if x has shape parameter μ and scale parameter ω, then x2 has a gamma distribution with shape parameter μ and scale parameter ω/μ.

## Nakagami-q Distribution / Hoyt Distribution

If the shape parameter is restricted to the interval q=μ; 0 > q > 1, the distribution is called Nakagami-q, also known as the Hoyt distribution or Nakagami-Hoyt distribution. Like the Nakagami-w, the scale parameter must be positive and the tails are relatively thin. Close to the mean, the Nakagami distribution is a good approximation to the Rician distribution. However, towards the tails they differ significantly.

Reference:
Nakagami, M. (1960) “The m-Distribution, a general formula of intensity of rapid fading”. In William C. Hoffman, editor, Statistical Methods in Radio Wave Propagation: Proceedings of a Symposium held June 18–20, 1958, pp 3-36. Pergamon Press.

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