Normal

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What does it mean to be normal?

The Cumulative Distribution Function, CDF, of an exponential distribution is an ideal function which has a value of zero at 0 and a value of 1 at infinity. However it is NOT a normal distribution, in that it has a skew of 2, as opposed to a normal distribution which has a skew of 0. An Exponentially Modified Gaussian Distribution has been proposed which combines the exponential and Gaussian (normal) distributions, but which results in a CDF with three parameters: µ, the mean; σ, the standard deviation; and λ, the rate parameter of the exponential distribution. A Gaussian distribution is NOT the only normal distribution, with a skew of 0 and where the mean, median and the mode are equal. Another normal distribution is the Logistics distribution which has two parameters: µ, the mean; and s, the scale parameter. However since s is a constant factor of of σ, s=√3/π * σ, it effectively has the same parameters as the Gaussian distribution. Reyes et al (Reyes, Venegas, & Gómez, 2018) proposed an Exponentially Modified Logistic distribution which has only two parameters : µ, the mean; and s, the scale parameter where the rate parameter of the exponential distribution is also equal to s.  However it is a solution where s, which is a constant factor of σ, is equal to λ, the rate parameter of an exponential distribution. It could have just as easily solved by setting µ equal to λ.

Figure 1 shows the Cumulative Distribution function, CDF, when σ, µ, and λ are all equal, in this case to 1. The CDF of the exponential function has the desired properties of being 0 when x is 0 and being 1 when x is infinity. Reyes’ Exponentially Modified Logistic distribution is closer to  normal distributions, whose CDFs are not zero when x is 0, but its CDF is closer to zero than that the CDF of a normal distribution such as a Gaussian or Logistic Distribution.

Figure 1 Cumulative Distribution functions, CDFs where µ=σ=λ

Figure 2 shows the CDFs when µ and λ are equal at 0.5, but the standard deviation σ, is twice that amount at 1.0. Also shown is the Exponentially Modified Gaussian CDF if the mean of an exponential distribution, 1/λ, is the equal to the mean of the Gaussian distribution, µ, as well as the Exponentially Modified Gaussian distribution if the standard deviation of an exponential distribution, 1/λ, is equal to the standard deviation of a Gaussian distribution, σ.

Figure 2 Cumulative Distribution functions, CDFs where µ=λ and σ=2*µ

Figure 3 shows the CDFs when σ and λ are equal at 0.5, but the mean µ is twice that amount at 1.0. Also shown is the Exponentially Modified Gaussian distribution CDF if the mean of an exponential distribution, 1/λ, is the equal to the mean of the Gaussian distribution, and the Exponentially Modified Gaussian distribution CDF if the standard deviation of an exponential distribution, 1/λ, is the equal to the standard deviation of the Gaussian distribution, σ.

Figure 3 Cumulative Distribution functions, CDFs where µ=λ and σ=2*µ

In all three figures, Reyes’ Exponentially Modified Logistics Distribution is closer to the shape of the CDF of normal distributions than the Exponentially Modified Gaussian distribution. When the parameters of the distributions are not equal, then the Exponentially Modified Gaussian Distribution does has a lower CDF when x is 0, but its does so by being less “normal” and closer to the shape of a skewed exponential distribution.

If a distribution is expected to be normal, then for its CDF to be close to the ideal function, it is suggested that Reyes’ Exponentially Modified Logistics Distribution be used. For this function to be normal, it appears to be more important that the standard deviation, σ, be equal to the inverse of the rate parameter of an exponential distribution, λ, than for the mean, µ, to be equal to the inverse of the rate parameter of an exponential distribution, λ.

 Reyes, J., Venegas, O., & Gómez, H. W. (2018). Exponentially-modified logistic distribution with application to mining and nutrition data. Appl. Math 12.6, 1109-1116.