Before comparing and contrasting discrete and continuous probability distributions, it is vital to understand what discrete variables and continuous variables are. A discrete variable is a variable that has countable values, for example, a list of non-negative integers. On the other hand, a continuous variable is a variable with a set of possible values (often referred to as range) that is infinite and uncountable. Thus, if a variable can take a specific range of values, it is called a continuous variable; otherwise, it is called a discrete variable.
Depending on the variables they define, all probability distributions can be categorized into two broad categories: discrete probability distributions (DPDs) or continuous probability distributions (CPDs). DPDs define probabilities associated with discrete variables, whereas CPDs define probabilities associated with continuous variables (Anderson et al., 2016). With a DPD, each discrete variable can be associated with a non-zero probability. Therefore, a DPD can always be presented in tabular form (Anderson et al., 2016). The probability distribution of discrete random variables is often called probability mass distribution (PMF). On the other hand, the probability that a continuous random variable will assume a particular value is zero. Thus, a CPD cannot be expressed in tabular form. Instead, CPDs are often expressed using equations or formulas (Anderson et al., 2016). In most cases, the equation used to describe CPDs is called a probability density function (PDF). Thus, CPDs are normally described in terms of PDF.
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DPDs and CPDs have one thing in common. All discrete and continuous variables have a cumulative distribution function (CDF). Thus, DPDs and CPDs can always be presented using CDF. This is because there is always a CDF for discrete random variables as well as a CDF for continuous random variables.
References
Anderson, D. R., Sweeney, D. J., Williams, T. A., Camm, J. D., & Cochran, J. J. (2016). Statistics for business & economics . Nelson Education.
