Probability theory is one of the most important concepts in statistics. It can be said that it is an approach to statistical analysis that attempts to give an explanation as to the probability of a specific event occurring. Probability is used in many scientific disciplines, including economics, physics, actuarial science and engineering, and in many industries such as marketing, investment banking and telecommunications. There exists a vast number of well-known probability distributions, which have been developed over the years. One of the most famous is the binomial distribution, which has been developed in the 1940s.
The binomial distribution comes in two forms: the normal and binomial normal. In the normal distribution, the probability of a particular event does not depend on the value of one parameter, while in the binomial normal, the probability depends on the value of at least two parameters. Probability of a normal distribution is often compared to the probability density function, which is often referred to as the log-normal distribution, because it has the same shape as the log function (i.e., the value tends to be just half log(x) as the square of the average value of the input variable). For example, if we look at the log-normal distribution, the probability density function will tend to have values close to unity (i.e., a value near zero on the log function). Similarly, the binomial, normal distribution has distributions that have values close to zero (zero on all the inputs) and ranges that encompass the range that is the mean of the log-normal values.
The binomial distribution is one of the more complicated distributions to understand, and many students find difficulty in applying the theoretical distributions to real data. Binomial normal distributions are characterized by their mean (or mean) function that tends to be zero over the interval of the distribution, as well as mean values that are close to unity. These distributions are useful for analyzing any number of financial problems, such as investment portfolios, portfolio diversification, and portfolio optimization. However, their use in this context may be limited due to the high level of complexity associated with their design. This is why it’s often necessary to learn more about the underlying probability distribution prior to working with binomial, normal distributions themselves.
The standard deviation is a simpler but still highly important, model in the analysis of probability distributions. Standard deviation is based on the deviation of the mean value of a set of random variables from the mean value, that they are supposed to be drawn from by an equally distributed random variable. The deviation of the mean value can be thought of as the deviation of the arithmetic mean from the normal distribution itself. Standard deviation maps the expected range of a random variable to its real range, so that deviations of the mean from the mean are termed’standard deviations’. Because there are no true limits to the range of deviation, the standard deviation allows for a range over a given range of values that are continuous.
Another important theoretical distribution used in many financial risk management analyses is the log-normal distribution. The log normal distribution follows a binomial curve, with mean and standard deviation equal to one and the range of values between the two being equal to one minus the square root of the number being grouped under the binomial curve. It is this range that can be used as a measure of risk, since the log-normal distribution follows a binomial curve that implies the probability of tails of outcomes being equally likely. Important uses of the log-normal distribution in risk management are in the option-based valuation of stock portfolio risks and in calculating the optimal level of return. Since the log-normal distribution is based on log-normal data, its results are also valid for other normal data sets, including non-normal data sets.
One of the most widely used statistical distributions is the logistic curve. Also known as the Kauffman-Krueger curve, the logistic distribution is often implemented in many economic models because of its ability to fit a wide range of non-normal data sets. The key advantage of the digitization procedure is that it allows multiple points on the curve to be compared at any time, which greatly increases the ability to represent complex or illogical data. The mathematical forms of these distributions can be quite complicated and are often used to fit the volatility, riskiness, breakeven, and other parameters of financial risk management scenarios.
A third popular statistical distribution used in many risk management applications is the logistic normal distribution, which is derived from the logistic function by taking the log-normal values of all possible outcomes, which can include tails of outcomes. Like the log-normal distribution, the logistic curve can be used to fit many different non-normal data sets. When a model is fitted using a logistic curve, the range of distributions can be anything between zero (absent tails) to infinity (including full tails). The mathematical forms of these distributions are closely related to the mathematical forms used in calculus, but they are often easier to learn and use.
While these three distribution types are excellent tools for providing insights into the range of possible results with data analysis, they all assume a prior probability distribution, which is a necessary component in all cases. Frequent methods of distributing likelihoods include binomial, Poisson, and so on. While the method of choice may be different for each application, the range of probabilities that can be achieved by using these methods are essentially the same.