Probability Distribution

- Overview

A probability distribution is a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range. 

This range will be bounded between the minimum and maximum possible values, but precisely where the possible value is likely to be plotted on the probability distribution depends on a number of factors. 

A probability distribution depicts the expected outcomes of possible values for a given data-generating process. Probability distributions come in many shapes with different characteristics, as defined by the mean, standard deviation, skewness, and kurtosis.

Investors use probability distributions to anticipate returns on assets such as stocks over time and to hedge their risk.

Please refer to the following for more information:

• Wikipedia: Probability Distribution 

- Deterministic Experiments

In probability theory, a deterministic experiment is a procedure that has only one possible outcome. The same initial conditions and rules will always produce the same outcome. For example, calculating the product of 2 multiplied by 3 is a deterministic experiment. 

In contrast, a random experiment has more than one possible outcome. The outcome of a random experiment is determined by chance. 

The experiments which have only one possible result or outcome i.e. whose result is certain or unique are called deterministic or predictable experiments. The result of these experiments is predictable with certainty and is known prior to its conduct. 

- Non-Deterministic Experiments

In probability theory, a non-deterministic experiment is a procedure that can be repeated infinitely and has a well-defined set of possible outcomes. 

The outcomes of a non-deterministic experiment are not determined by the initial conditions and rules. Instead, the outcome is influenced by chance or randomness. 

Here are some examples of non-deterministic experiments: 

• Rolling a fair die
• Calculating your savings account balance in a month
• The relationship between a circle's circumference and radius, or the area and radius of a circle
• Tossing a coin

- The Sample Space

Sample Space is a concept in probability theory that deals with the likelihood of different outcomes occurring in a given experiment. It involves defining a sample space that encompasses all possible outcomes and assigning probabilities to these outcomes. 

In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. 

A sample space is usually denoted using set notation, and the possible ordered outcomes, or sample points, are listed as elements in the set. It is common to refer to a sample space by the labels S, Ω, or U (for "universal set"). The elements of a sample space may be numbers, words, letters, or symbols. They can also be finite, countably infinite, or uncountably infinite. 

- Probability Distributions

A probability distribution is a mathematical function that describes the probabilities of different outcomes for an experiment. It's a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events. 

Here are some requirements for a probability distribution: 

• The value of the random variable X must be described.
• All the values of random variable X must have their corresponding probabilities.
• The sum of all probabilities is equal to one.

- Types of Probability Distributions 

There are many different classifications of probability distributions. Some of them include the normal distribution, chi-square distribution, binomial distribution, and Poisson distribution. The different probability distributions serve different purposes and represent different data generation processes.

Here are some types of probability distributions: 

• Binomial distribution: Models the probability of obtaining one of two outcomes, a certain number of times (k), out of a fixed number of trials (N) of a discrete random event.
• Poisson distribution: Models the probability that a given number of events will occur within an interval of time independently and at a constant mean rate.
• Bernoulli distribution: Describes a coin flip. There are two possible outcomes, 0 and 1 (equivalently, False and True).
• Normal distribution: A type of probability distribution where the mean, the median, and the mode are all equal to each other.
• Geometric distribution: Represents the probability of getting a number of successive failures till the first success is obtained.
• Uniform distribution: A type of probability distribution in which all outcomes are equally likely.
• Exponential distribution: Describes the time between events in a Poisson process, where events occur continuously and independently at a constant average rate.
• Gamma distribution: A continuous probability distribution that is typically used to model positive-valued variables with skewed distributions. 

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- How Probability Distribution Works

Perhaps the most common probability distribution is the normal distribution or "bell curve," although several commonly used distributions exist. Often, the data generation process for some phenomenon will determine its probability distribution. This process is called the probability density function. 

Probability distributions can also be used to build a cumulative distribution function (CDF), which accumulates the probability of occurrence and always starts at 0 and ends at 100%. 

Academics, financial analysts, and fund managers alike can determine the probability distribution of a particular stock to assess the expected returns that stock may generate in the future. 

A stock's return history can be measured from any time interval, but is likely to include only a small portion of a stock's returns, which will subject the analysis to sampling error. By increasing the sample size, this error can be significantly reduced.

- The Formula for Probability Distribuntion

The probability distribution function is written as F(x) = P (X ≤ x). For a semi-closed interval (a, b], the probability distribution function is P(a < X ≤ b) = F(b) - F(a). The probability distribution function of a random variable is always between 0 and 1. 

The probability distribution of X is the function f(x) p(x)= P(X=x) for each x within the range of X. It is often called the probability mass function for the discrete random variable X. 

The formula for the discrete probability distribution is X ∼ G(p). The formula for the pmf
 is P(X = x) = (1 - p)x p, where p is the success probability of the trial. 

The normal distribution formula for a random variable x, with mean “μ” and standard deviation “σ”, is f(x) = (1/√(2πσ2)) (e[-(x-μ)^2]/2σ^2).

- Example: Probability Distributions and Frequency Distribution

A probability distribution is a mathematical function that describes the probability of different possible values of a variable. Probability distributions are often depicted using graphs or probability tables.

A probability distribution is an idealized frequency distribution. A frequency distribution describes a specific sample or data set. It is the number of times each possible value of a variable appears in the data set. 

The number of times a value appears in a sample is determined by its probability of occurring. A probability is a number between 0 and 1 that represents the likelihood of something occurring: 0 means it is impossible. 1 means it is certain.

The higher the probability of a value, the higher its frequency in the sample. More specifically, the probability of a value is its relative frequency in an infinitely large sample. 

Infinitely large samples are not possible in real life, so probability distributions are theoretical. They are idealized versions of frequency distributions that are meant to describe the population from which the sample came. 

Probability distributions are used to describe populations of real-life variables, such as coin tosses or the weight of an egg. They are also used in hypothesis testing to determine p-values.

Example: Suppose a chicken farmer wants to know the probability that the eggs produced on her/his farm are a certain size. The chicken farmer weighs 100 random eggs and describes their frequency distribution using a histogram.

She/his can get a rough idea of ​​the probabilities of different egg sizes directly from this frequency distribution. For example, she can see that the probability of an egg being approximately 1.9 ounces is high, while the probability of an egg being larger than 2.1 ounces is low.

Suppose the chicken farmer wants to estimate the probabilities more precisely. One option is to improve her estimate by weighing more eggs.

A better option is to realize that egg sizes seem to follow a common probability distribution called a normal distribution. The chicken farmer can assume that egg weights are normally distributed, resulting in an idealized distribution of egg weights.

Because statisticians understand the normal distribution very well, the chicken farmer can calculate precise probability estimates even with relatively small sample sizes.

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