A Random Variable is any rule that maps (links) a number with each outcome in sample space S. Mathematically, random variable is a function with Sample Space as the domain. It’s range is the set of Real Numbers.
In the Negative Binomial Distribution, we are interested in the number of Failures in n number of trials. This is why the prefix “Negative” is there. When we are interested only in finding number of trials that is required for a single success, we called it a Geometric Distribution.
Binomial Distribution is used to find probabilities related to Dichotomous Population. It can be applied to a Binomial Experiment where it can result in only two outcomes. Success or Failure. In Binomial Experiments, we are interested in the number of Successes.
Probability Mass Function (PMF) of X says how the total probability of 1 is distributed (allocated to) among the various possible X values.