Q. How do I find the probabilities of the different values assumed by random variables?
There are two different types of functions used to mathematically find probabilities: probability mass/density functions and cumulative density functions.
Probability functions are defined mathematically as \(f_X(x) = P(X = x)\), where X is a random variable and x is a real number. Probability mass functions (pmfs) are used for discrete random variables and probability density functions (pdfs) are used for continuous random variables. This distinction is made because discrete and continuous random variables have different ranges.
Cumulative density functions are defined mathematically as \(F_X(x) = P(X \leq x)\), where X is a random variable and x is a real number). The CDF can be found by integrating the pdf of a continuous random variable, or by summing the pmf of a discrete random variable for all values less than or equal to x. The pdf/pmf is also the derivative of the CDF.