Consider a particle that executes a periodic path over the interval [0, T] given by

x = x(t) for t ∈ [0, T] and x(T)=x(0)

The velocity v of the particle also goes through a period path given by

v = (dx/dt) = v(t) for t ∈ [0, T] and v(T)=v(0)

The same would apply for any higher derivative of x(t) but x and v are the primary concern now.

The Time-Spent Probability Density Functions

The probability that the particle would be found in an interval of length Δx at a randomly chosen t is proportional to the time the particle spends in that interval. That time is Δx/|v|, where v is the average velocity of the particle in that interval. As Δx goes down to an infinitesimal dx the time spent goes to dx/|v|. Thus 1/(T|v|), where T is the period of the motion, is the probability density for the particle.

Likewise the time-spent probability density function for velocity is given by 1/(T|(dv/dt)|=1/|a|, where a is acceleration. T is the time period for the particle motion.

T = ∫dt = ∫dx/|v| = ∫dv/|(dv/dt)|

More generally one can say

T = ∫dt = ∫dz/|dz/dt|

where z is the n-th order derivative of x(t), (dⁿx/dtⁿ), including the zeroeth order derivative which is x(t) itself.

Proof:

It is merely a matter of a change in the variable of integration for the integral ∫dt. If t is expressed as a function of z, t(z) then

∫dt = ∫(dt/dz)dz
but (dt/dz) = 1/(dz/dt)
hence
∫dt = ∫dz/(dz/dt)

The new variable of integration can be anything including x and the derivatives of x with respect to t.

Since

T = ∫dt = ∫dz/(dz/dt)
it follows that
1 = ∫dt/T = ∫dz/[T(dz/dt)]

and thus 1/[T(dz/dt)] is the probability density distribution function for z.

The Significance of Time-Spent
Probability Distributions

The expected value of the effects of a particle with charge q executing a periodic path x(t) is the same as if the charge q were smeared over the path with a distribution equal to the probability density function 1/(Tv(x)). The charge may be mass (gravitational charge) as well as electrostatic charge. For on this see Static/Dynamic. The time-spent probability density function for a an object gives its dynamic appearance as though it is a static, translucent object like a rapidly rotating fan appearing as a blurred disk.