Theoretical Distributions

Introduction:

In daily life and practical scenarios, there is rarely any experiment with one definite outcome like on or off, or a particular thing will happen or will not happen. In statistics, specific procedures are repeated for n number of times, and the results are accumulated and represented as theoretical distribution. 

The frequency distribution is obtained by applying a mathematical model, mean is taken in the theoretical distribution. In other words, theoretical distribution is a statistical distribution received by a set of logical and mathematical reasoning from given principles or assumptions. Theoretical distribution is the opposite of distribution derived by real-world data derived by empirical research. Empirical research distribution is compared with a suitable theoretical distribution to predict the probability of an event in the future. Relation between both the results can predict the probability of an event in the future.

Division of Theoretical Distributions

Theoretical distributions can be divided into Discrete Distributions and Continuous Distributions. Let us learn about these categories. 

Discrete distributions can be further divided into these categories;

• Binominal Distribution
• Poisson Distribution
• Expected Frequency or Normal Distribution

Binominal Distribution:

Binominal distribution deals in only two outcomes of a process in a given period. For example, the head or tails of a coin flip or the success or failure of an event. It only applies to discrete random variables. The binomial distribution summed up the number of trials, surveys, or experiments. It is advantageous when each option has an equal chance of achieving a specific value. Binominal distribution is based on the assumption that there is only one outcome of an event and that this outcome has an equal chance of occurring. Binominal distribution has three essential criteria: a fixed number of experiments or trials, probability of outcome stays same or equivalent, and every trial is independent of previous or succeeding trial or process.

Poisson Distribution:

Poisson distribution helps predict the future occurrence of a constant and continuous event in circumstance.

For example, Poisson random variable will give flu cases in a particular age group in a given geographical area in a given period. The distribution will provide a range of probability from zero to infinity. The number of successes from a Poisson experiment is called Poisson random variables. Poisson distribution has four basic properties; the outcome is countable in the whole number. The average frequency of a product in a particular time interval is known. The probability of the result of an event is not dependent on previous occasions. The probability of more than one outcome in a given time interval is low. 

Expected Frequency or Normal Distribution

Random variables that follow the normal distribution have values that can take on any known value within a specific range. With mean and variance, binomial distribution changes into a normal distribution. The probability of a given value in a specified field in a particular process or trial is obtained by a normal distribution. The normal distribution has these properties; it has a symmetrical shape; the mean and median are the same value and are located in the center of the distribution, its standard deviation measures the distance on distribution between the mean and the inflection point; the Empirical Rule is used to calculate the probabilities for the normal distribution and its values are almost all within three standard deviations of the mean.

Continuous Distributions:

The continuous uniform distribution is the most basic probability distribution; however, it is very effective in modeling random variables; hence it is essential in statistics. It is a continuous uniform distribution, which means that it accepts values between a specific range, such as 0 and 1. The uniform distribution is a constant probability distribution, and it is concerned with occurrences with an equal chance of occurring. The continuous random variable is said to have a uniform distribution or a rectangular distribution on the interval.

Continuous distribution can be further divided into these categories;

• Normal distribution
• Exponential distribution

Normal Distribution:

The essential distribution for characterizing a continuous random variable is the normal probability distribution.

The normal distribution is the most commonly assumed form of distribution in technical and other statistical investigations. The mean and standard deviation are the two parameters of the standard normal distribution. A normal distribution has 95 percent within two standard deviations, 68 percent of the impression, and 99.7 percent within three.

The Central Limit Theorem motivates the normal distribution model. According to this theory, regardless of the type of finite variance distribution from which the variables are selected, averages calculated from independent, identically distributed random variables have nearly normal distributions.

The normal distribution, sometimes also referred to as the Gaussian distribution, is symmetric about the mean probability distribution, indicating that data closer to the mean occur more frequently than data further away from the mean. The normal distribution will show as a bell curve on a graph.

Exponential Distribution

The exponential probability distribution can describe how long it takes to finish an activity. The exponential distribution is generally used to predict when an event will occur.

In the field of reliability, the exponential distribution is commonly utilized. It is inextricably linked to the Poisson model. If failures happen in a Poisson model, the time between successive defeats is exponential.

This model can also predict the chance of a given number of defaults occurring during a given period. It is frequently used to simulate the time spent between occurrences.

Conclusion:

Discrete and continuous random variables are not interchangeable as the value of a discrete variable is derived by counting, e.g., the number of females in a particular area. The value of a continuous variable is derived by measurement, e.g., children’s weight in a park. A random variable is the numerical value of an outcome of an unexpected event. In short, a discrete and continuous random variable means there is a countable and finite number of possible values.