pnormratio {gaussratiovegind} | R Documentation |
Cumulative Distribution of a Normal Ratio Distribution
Description
Cumulative distribution of the ratio of two independent Gaussian distributions.
Usage
pnormratio(z, bet, rho, delta)
Arguments
z |
length |
bet , rho , delta |
numeric values. The parameters |
Details
Let two independant random variables
X \sim N(\mu_x, \sigma_x)
and Y \sim N(\mu_y, \sigma_y)
.
If we denote \displaystyle{ f_Z(z; \beta, \rho, \delta_y)}
the probability distribution function of the ratio
\displaystyle{Z = \frac{X}{Y}}
,
with \beta = \frac{\mu_x}{\mu_y}
,
\displaystyle{\rho = \frac{\sigma_y}{\sigma_x}}
and \displaystyle{\delta_y = \frac{\sigma_y}{\mu_y}}
(see dnormratio()
, Details section).
The probability distribution for Z
is given by:
\displaystyle{F(z; \beta, \rho, \delta_y) = \int_{-\infty}^z{f_Z(z; \beta, \rho, \delta_y)}}
This integral is computed using numerical integration.
Value
Numeric: the value of density.
Author(s)
Pierre Santagostini, Angélina El Ghaziri, Nizar Bouhlel
References
El Ghaziri, A., Bouhlel, N., Sapoukhina, N., Rousseau, D., On the importance of non-Gaussianity in chlorophyll fluorescence imaging. Remote Sensing 15(2), 528 (2023). doi:10.3390/rs15020528
Marsaglia, G. 2006. Ratios of Normal Variables. Journal of Statistical Software 16. doi:10.18637/jss.v016.i04
Díaz-Francés, E., Rubio, F.J., On the existence of a normal approximation to the distribution of the ratio of two independent normal random variables. Stat Papers 54, 309–323 (2013). doi:10.1007/s00362-012-0429-2
See Also
dnormratio()
: density function.
rnormratio()
: sample simulation.
estparnormratio()
: parameter estimation.
Examples
# First example
beta1 <- 0.15
rho1 <- 5.75
delta1 <- 0.22
pnormratio(0, bet = beta1, rho = rho1, delta = delta1)
pnormratio(0.5, bet = beta1, rho = rho1, delta = delta1)
curve(pnormratio(x, bet = beta1, rho = rho1, delta = delta1), from = -0.1, to = 0.7)
# Second example
beta2 <- 2
rho2 <- 2
delta2 <- 2
pnormratio(0, bet = beta2, rho = rho2, delta = delta2)
pnormratio(0.5, bet = beta2, rho = rho2, delta = delta2)
curve(pnormratio(x, bet = beta2, rho = rho2, delta = delta2), from = -0.1, to = 0.7)