Planning

 

I Want To Be Happy

I'm a very ordinary man
Trying to work out life's happy plan
Doing unto others as I'd like to have them doing unto me

In life's happy  plan, are things random or deterministic?

By definition, a random number has two parameters: its location, μ, and its scale, σ. To define a deterministic number, the scale must be equal to zero, which is equivalent to saying that there is only one parameter, its location.

If the scale is non-zero, the system is random. If the scale is zero, the system is deterministic. Is the universe in which we exist random, or deterministic? By this it is meant the universe as a whole, not an object within that universe. The properties of an object within the universe can be deterministic even when the shape of the universe is random. The scientific standard for the properties of an object is 3 Sigma, 3σ, that is setting σ as close to zero as possible. This means that you are 99.7% certain that the location is the only value. In particle physics, an even more strict standard applies, 5 Sigma, 5σ. This means that you are 99.99994% certain that the location is the only value.

These percentages are due to the 68/95/99 rule of normal distributions. That is

·        68% of the observations are within 1 Standard Deviation, σ, of the mean, μ; that is μ ± σ,

·        95% of the observations are within 2 Standard Deviations, 2σ, of the mean, μ, that is μ ± 2σ,

·        99.7% of the observations are within 3 Standard Deviations, 3σ, of the mean, μ, that is μ ± 3σ.

There are distributions which have only one parameter. The most common of these is the Exponential Distribution. Its Probability Density Function, PDF, can be stated using a single parameter, λ, as

 λ*e^-λ*x for x>0 and x∈ R.

Its Cumulative Distribution Function, CDF, which is the integral of its PDF, is

1- e^-λ*x for x>0 and x∈ R.

Its mean is 1/λ. Its median is 1/λ.   Its variance, which is the square of the Standard Deviation, is 1/λ². Its skewness is 2. It does not meet the 68/95/99 requirement and is therefore not normal.

This can be translated on the x- axis from an origin of (0,0) to an origin of (μ,0). Then it has two parameters, λ and μ. Its PDF can be stated as y= λ*e^-λ*(x-μ) for x>μ and x∈ R. With these two parameters, the mean is μ+1/λ. Its median is μ+1/λ. But its variance remains 1/λ² and its skewness remains 2. And it still is not normal.

The Gaussian Normal Distribution also has two parameters, μ and σ. Its PDF is conventionally stated as

1/(σ√(2π)) e^{-½*((x-μ)/σ)} for x∈ R.

With these two parameters, the Gaussian Normal Distribution’s mean is μ. Its median is μ. Its variance, whose square is the Standard Deviation, is σ. Its skewness is 0.

The Exponential Distribution is often used in analyses where the location, µ, is known to be zero, for example, the average travel distance from the home in any direction. In this case the home is defined as at location = 0. Fitting the observations of the trip length with respect to the home to an Exponential Distribution gives the mean and median trip lengths.

However when the location is not zero, then an Exponential Distribution should not be used, for example, observations of the size of widgets that are being manufactured. In this case, you would like the observations to tell you the location and standard deviation of the observations and see if those meet the location and variance requirements for the size.

To combine the features of the exponential and random distributions, it has been proposed that there be an Exponentially Modified Gaussian distribution, EMG, which includes all three parameters, but the Probability Density Function and the Cumulative Distribution Function become very complex. The PDF is

λ/2 e^(λ/2 (2*μ+λ*σ^2-2x) ) erfc((μ+λ*σ^2-x)/(√2 *σ))

where erfc(x), the complementary error function, is 1-erf(x), or

(2/√π)* ∫_x^∞ e^(-(t^2) )

The Cumulative Distribution Function of the EMG, the EMG’s CDF, the integral of its PDF, is

Φ(x, μ, σ)-1/2 e^(λ/2 (2μ+λ*σ^2-2x) ) erfc((μ+λ* σ^2-x)/(√2* σ))

where Φ(x, μ, σ) is the is the CDF of a Gaussian distribution.

The mean of an EMG is μ+1/λ. Its median is the value of x at which the EMG’s CDF is 50%. The variance is σ²+1/λ². Its skewness is (2/(σ³λ³)) (1+1/(σ²λ²))^-3/2.  By inspection, the skewness takes on a value between 0.00 and 0.31.

The Gaussian Distribution is not the only Normal Distribution. The Logistics Distribution, also known as the Sech Squared, the square of the hyperbolic secant, Distribution, is a normal distribution which has a PDF of

(1/(4*s))* sech² ((x-μ)/(2*s))

where s is a scale parameter which is equal to √3/π *σ. Its CDF is

(1/2)+(1/2)*tanh((x-μ)/(2*s))

Its mean is μ. Its median is μ. Its variance is s²π²/3. Its skewness is zero.

The CDF of the Logistics Distribution is identical to a logit choice of 0 or 1 at a location of μ where the odds of making a choice of 1 are 50%. In this case the value of s is 0.5, 50%.

A doubling of the logit choice CDF is almost identical to the CDF of a translated exponential distribution. A Standard Normal Logit Choice Distribution is proposed to always have a
σ = √3/0.5π= .91 and have a PDF of sech²(√(√3/π) (x-μ)) for x>μ and x∈ R. This means that it has a CDF of tanh(√((√3)/0.5π) (x-μ)) for x>μ and x∈ R. 

The plot of an exponential association, which is the CDF a conventional exponential distribution, and, the CDF of a Standard Normal Logit Choice Distribution, where μ=0, is shown in Figure 1. The correlation between the two equations between x=0 and x=3 is 0.999. 

Figure 1 Exponential Association and  CDF of Standard Normal Logit Distribution

This suggests that the universe is random, and not deterministic, because the universe has a σ  = 0.91.