Continuous Probability Distributions

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Continuous probability distributions are needed when the random variable of interest can assume any value inside of one or more intervals of real numbers such as, for example, any number greater than zero. Asset returns, for example, whether measured monthly, weekly, daily, or at an even higher frequency are commonly modeled as continuous random variables. In contrast to discrete probability distributions that assign positive probability to certain discrete values, continuous probability distributions assign zero probability to any single real number. Instead, only entire intervals of real numbers can have positive probability such as, for example, the event that some asset return is not negative. For each continuous probability distribution, this necessitates the so-called probability density, a function that determines how the entire probability mass of one is distributed. The density often serves as the proxy for the respective probability distribution. Keywords: continuous distribution function; cumulative distribution function; cumulative probability distribution function; P-null sets; density function; marginal rate of growth; derivative; probability density function; density function; integral; support; continuous random variable; measurable function; state space; parametric distributions; mean; mean; moments of higher order; first; moment of order k; variance; tails; standard deviation; skewness; Pearson skewness; left skewed; right skewed