Final answer:
The question revolves around mixture distributions and their probabilities, which is a statistical concept within mathematics at the college level.
Step-by-step explanation:
The question appears to be focused on probability mass function (pmf) within the topic of mixture distributions, likely tied to the field of statistics which is a part of mathematics. A mixture distribution can be thought of as a probability distribution made up of several component distributions, typically when the population being studied is a mix of two or more subpopulations, each with its own distribution. In this case, the student seems to be referencing the distribution of two populations characterized by subscripts M for males and F for females. The reference to PM and PF suggests these are the proportion of each subpopulation within the mixture.
To assess or test these proportions, the expectation value of a random variable, which is the theoretical mean of the variable, might be considered to determine the overall distribution's behavior. Examples in the provided information like distributing particles among boxes and the calculation of microstates relate to statistical thermodynamics, which connects the concept of entropy, microstates, and probabilities in a closed system. The expectation value of the momentum or average momentum is another statistical concept that could relate to the general expectations of a distribution or mixtures of distributions within physics.