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I recently learned the concept of Maxwell's speed distribution and became interested in how to use similar momentum distributions to study the probabilistic motion of a classical free particle. I have done some of my own reading on probabilities and distributions (no formal lessons yet) and with some help, tried to derive a probability density for a classical particle. Could someone look through my work and point out mistakes in my understanding?

The classical equation of motion for a free particle is (##m = 1##),

$$X = x_0 + pt$$

where ##x_0## and ##pt## are random variables.

Let ##F## denote Cumulative Distribution Function and ##f## for Probability Density

$$F_X = \int_{-\infty}^{\infty} F_{pt} (a - y) f_{x_0}(y)dy$$

$$f_X = \int_{-\infty}^{\infty} f_{pt} (a - y) f_{x_0}(y)dy$$

Since this is a classical particle we talk about, then ##f_{x_0}(y) = \delta (y - x_0)## since it has ##P = 1## of being at ##x_0##

This gives,

$$f_X = f_{pt} (a - x_0) $$

If I know the momentum probability density ##f_p##, I could use the fact that

$$F_p = P(p<a) = \int_{-\infty}^{a} f_p(b) db$$

$$F_{pt} = P(pt<a) = P(p<\frac{a}{t}) = \int_{-\infty}^{\frac{a}{t}} f_p(c) dc$$

and do a quick change of variables, compare integrands (both with limits ##-\infty## to ##a##) that would give,

$$f_{pt}(b) = \frac{1}{t} f_p(b/t)$$

Finally,

$$f_X = f_{pt} (a - x_0) = \frac{1}{t} f_p( \frac{a - x_0}{t})$$

Which gives me a probability density of a free particle , ##f_X##

Does the above work? Many thanks in advance!