Distribution of the Ratio of Normal and Rice Random Variables

Abstract

The ratio of independent random variables arises in many applied problems. The distribution of the ratio X Y is studied when X and Y are independent Normal and Rice random variables, respectively. Ratios of such random variables have extensive applications in the analysis of noises in communication systems. The exact forms of probability density function (PDF), cumulative distribution function (CDF) and the existing moments are derived in terms of several special functions. As a special case, the PDF and CDF of the ratio of independent standard Normal and Rayleigh random variables have been obtained. Tabulations of associated percentage points and a computer program for generating tabulations are also given.

Improved methodology and set-point design for diagnosis of model-plant mismatch in control loops using plant-model ratio

“…As a result, |Ŷ p (ω n )| and |Ŷ m (ω n )|, being square-roots of the sum of squares of two normal random variables, follow Rice distributions.M(ω n ) is then a ratio of two Rician random variables. This distribution has been studied by Khoolenjani and Khorshidian [17]. However, at ω n = 0, Im(Ŷ p (ω n )) and Im(Ŷ m (ω n )) are equal to zero.…”

“…The magnitude of PMR is thus a ratio of two correlated Rice random variables. Khoolenjani and Khorshidian [17] have derived the probability density function for the ratio of two independent Rice random variables. The probability density function is difficult to derive for the ratio of correlated Rice variables.…”

“…The probability density function is difficult to derive for the ratio of correlated Rice variables. However, by using the Taylor expansions and the delta method [17], we can obtain approximate mean and variance of the magnitude of PMR: The approximate means and variances of |Ŷ p (ω n )| and |Ŷ m (ω n )| are given in the equations below. Here, 1 F 1 (a ; b ; x) denotes confluent hypergeometric function of the first kind: Similar results apply for |Ŷ m (ω n )|.…”

Secrecy Performance Analysis of Half/Full Duplex AF/DF Relaying in NOMA Systems Over κ - μ Fading Channels

“…is the modified Bessel function of the first kind and µ − 1 order. Using the series expansion of the modified Bessel function of the first kind in [41] given as…”

“…Using the series expansion of the marcum Q function in Eq(4.47) [38] and the definition of the power series in Eq(1.111) [41], Φ FD,∆ 2 can be expressed as where B FD 2 = A R,E + A R,1 C FD 1 and A R,1 = (1+κ)μ γ R,1 . Let t =…”