# Probability Density Function of the Eta-Mu Distribution

### Description

This function computes the probability density of the Eta-Mu (η:μ) distribution given parameters (η and μ) computed by paremu. The probability density function is

f(x) = \frac{4√{π}μ^{μ - 1/2}h^μ}{γ(μ)H^{μ - 1/2}}\,x^{2μ}\,\exp(-2μ h x^2)\,I_{μ-1/2}(2μ H x^2)\mbox{,}

where f(x) is the nonexceedance probability for quantile x, and the modified Bessel function of the first kind is I_k(x), and the h and H are

h = \frac{1}{1-η^2}\mbox{,}

and

H = \frac{η}{1-η^2}\mbox{,}

for “Format 2” as described by Yacoub (2007). This format is exclusively used in the algorithms of the lmomco package.

If μ=1, then the Rice distribution results, although pdfrice is not used. If κ \rightarrow 0, then the exact Nakagami-m density function results with a close relation to the Rayleigh distribution.

Define m as

m = 2μ\biggl[1 + {\biggr(\frac{H}{h}\biggl)}^2 \biggr]\mbox{,}

where for a given m, the parameter μ must lie in the range

m/2 ≤ μ ≤ m\mbox{.}

The I_k(x) for real x > 0 and noninteger k is

I_k(x) = \frac{1}{π} \int_0^π \exp(x\cos(θ)) \cos(k θ)\; \mathrm{d}θ - \frac{\sin(kπ)}{π}\int_0^∞ \exp(-x \mathrm{cosh}(t) - kt)\; \mathrm{d}t\mbox{.}

### Usage

 1 pdfemu(x, para, paracheck=TRUE)

### Arguments

 x A real value vector. para The parameters from paremu or vec2par. paracheck A logical controlling whether the parameters and checked for validity.

### Value

Probability density (f) for x.

W.H. Asquith

### References

Yacoub, M.D., 2007, The kappa-mu distribution and the eta-mu distribution: IEEE Antennas and Propagation Magazine, v. 49, no. 1, pp. 68–81