The cumulative distribution function (CDF) is a fundamental concept in probability theory. It describes the probability that a random variable takes a value less than or equal to a particular number. In other words, the CDF of a random variable X is defined as F(x) = P(X<x), where x is any real number.
The CDF provides a complete characterization of the distribution of a random variable. It is particularly important for continuous random variables because it enables us to calculate the probability that a random variable takes on a value within a certain interval. For example, suppose we have a continuous random variable X that represents the heights of adult males in a certain population. The CDF F(x) for X specifies the probability that a randomly selected male is less than or equal to x inches tall. Using this information, we can calculate the probability that a randomly selected male is between a certain range of heights, such as between 5’8" (68 inches) and 6’2" (74 inches).
The CDF is also useful for identifying percentiles of a distribution. For example, the 90th percentile of X represents the point at which 90% of the population has heights less than or equal to that value.
In quantitative trading and investment, the CDF is commonly used to model the probability of large price movements in financial markets. The CDF of daily returns can help traders and investors understand the risk and potential payoff of different investment strategies. For example, a trader may use the CDF to calculate the probability of experiencing a large loss or gain on a particular day, and adjust their trading strategy accordingly.