Short answers, pulled from the story.
A random variable is a mathematical function where the domain is the set of possible outcomes in a sample space and the range is a measurable space. This definition emerged from the rigorous axiomatic setup of measure theory to allow mathematicians to analyze chance without philosophical debates about uncertainty.
Pafnuty Chebyshev was the first person to think systematically in terms of random variables. He established a framework that treats a random phenomenon as a measurable function mapping outcomes to real numbers.
Discrete random variables take values in a countable subset and are described by a probability mass function. Continuous random variables take values in an interval of real numbers where the probability of selecting any single real number is zero and probability is assigned to intervals.
The probability distribution of a random variable is often characterized by moments which provide a practical interpretation of the data. The first moment represents the expected value or average value, while the variance and standard deviation answer how far from this average the values typically are.
Two random variables are equal in distribution if they have the same distribution functions but need not be defined on the same probability space. Two variables are equal almost surely if the probability that they are different is zero, a condition that is as strong as actual equality for all practical purposes.