The Maxwell distribution of speeds, f(v), is the starting point for the calculation of the transport coefficients in kinetic-molecular theory. Most physical chemistry textbooks follow a path to derive f(v) similar to that used by Maxwell, which makes it difficult for students to understand its relationship with the equilibrium state of the system, and its probabilistic character. The mathematical implications of the distribution, such as its relationship with other probability distribution functions, are also not sufficiently developed. Herein, we discuss how deriving the Maxwell distribution of speeds from the Boltzmann distribution function allows students to connect with the basic concepts of statistical thermodynamics, such as those treated in an introductory undergraduate course. We also present the relationship between Maxwell's distribution and other probability distribution functions, which can be helpful for deriving simple proportionality relationships between f(v) and v, under different conditions.