Perhaps the most common probability distribution is the normal distribution, or " bell curve ," although several distributions exist that are commonly used. Typically, the data generating process of some phenomenon will dictate its probability distribution. This process is called the probability density function. Academics, financial analysts and fund managers alike may determine a particular stock's probability distribution to evaluate the possible expected returns that the stock may yield in the future.
The stock's history of returns, which can be measured from any time interval, will likely be composed of only a fraction of the stock's returns, which will subject the analysis to sampling error. By increasing the sample size, this error can be dramatically reduced.
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. The binomial distribution, for example, evaluates the probability of an event occurring several times over a given number of trials and given the event's probability in each trial. Another typical example would be to use a fair coin and figuring the probability of that coin coming up heads in 10 straight flips.
A binomial distribution is discrete , as opposed to continuous, since only 1 or 0 is a valid response. The most commonly used distribution is the normal distribution, which is used frequently in finance, investing, science, and engineering. The normal distribution is fully characterized by its mean and standard deviation, meaning the distribution is not skewed and does exhibit kurtosis. This makes the distribution symmetric and it is depicted as a bell-shaped curve when plotted.
A normal distribution is defined by a mean average of zero and a standard deviation of 1. Unlike the binomial distribution, the normal distribution is continuous, meaning that all possible values are represented as opposed to just 0 and 1 with nothing in between. Stock returns are often assumed to be normally distributed but in reality, they exhibit kurtosis with large negative and positive returns seeming to occur more than would be predicted by a normal distribution.
In fact, because stock prices are bounded by zero but offer a potential unlimited upside, the distribution of stock returns has been described as log-normal. This shows up on a plot of stock returns with the tails of the distribution having greater thickness. Probability distributions are often used in risk management as well to evaluate the probability and amount of losses that an investment portfolio would incur based on a distribution of historical returns.
One popular risk management metric used in investing is value-at-risk VaR. VaR yields the minimum loss that can occur given a probability and time frame for a portfolio.
Alternatively, an investor can get a probability of loss for an amount of loss and time frame using VaR. Misuse and overeliance on VaR has been implicated as one of the major causes of the financial crisis. As a simple example of a probability distribution, let us look at the number observed when rolling two standard six-sided dice.
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Select personalised ads. Apply market research to generate audience insights. The distribution shows the family of probabilities and is a suitable model to depict random behaviour of percentages or proportions. It is used for the data models that hold uncertainties of the success probabilities in a random experiment. The general formulation of beta distribution is also known as the beta distribution of first kind and beta distribution of second kind is another name of beta prime distribution.
Beta distribution has many applications in statistical description of allele frequencies in genetic population, time allocation in project management, sunshine data, proportions of minerals in rocks, etc. Referred blog: Conditional Probability. Being the simplest form of Bayesian mode, beta-binomial distribution has extensive applications in intelligence testing, epidemiology, and marketing. The parametric shape can be defined in the form of the probability of success such that.
Talking about the key difference amid a beta-distribution and binomial distribution, the success probability, p, is always fixed for a set of trials whereas it is not fixed for beta-binomial distribution and changes trail to trail.
It is employed to estimate population parameters when the sample size is small, and the standard deviation is unknown. It is widely used for hypothesis testing and built confidence intervals for mean values. The graph of t-distribution distribution is shown below;. Similar to normal distribution, the t-distribution has bell-shaped curve distribution and is symmetric when mean is zero. As the sample size, n, increases, t-distribution acts as normal distribution where the considered sample size is greater than Must check: T-test vs Z-test.
Uniform distribution can either be discrete or continuous where each event is equally likely to occur. It has a constant probability constructing a rectangular distribution.
A variable X is said to have uniform distribution if the probability density function is. Must check: 4 types of data in statistics.
To sum up, we have seen various types of statistical data distribution models along with their probability density distribution functions, graphical representations and common properties. Be a part of our Instagram community. Introduction Definition of Statistics: The science of producing unreliable facts from reliable figures. Each trial has only two possible outcomes-success or failure.
Each event must be independent of each other. Read also: ANOVA test Binomial Distribution The binomial distribution is applied in binary outcomes events where the probability of success is equal to the probability of failure in all the successive trials. The two parameters are; The number of times an event occurs, n, and Assigned probability, p, to one of the two classes For n number of trials, and success probability, p, the probability of successful event x within n trials can be determined by the following formula The graph of binomial distribution is shown below when the probability of success is equal to probability of failure.
Binomial distribution The binomial distribution holds the following properties; For multiple trials provided, each trial is independent to each other, i. Normal Gaussian Distribution Being a continuous distribution, the normal distribution is most commonly used in data science. Normal distribution Normal distribution has the following properties; Mean, mode and median coincide with each other.
The distribution has a bell-shaped distribution curve. The distribution curve is symmetrical to the centre. The area under the curve is equal to 1. Recommended blog: Types of statistical Analysis Poisson Distribution Being a part of discrete probability distribution, poisson distribution outlines the probability for a given number of events that take place in a fixed time period or space, or particularized intervals such as distance, area, volume.
Poisson distribution considers following assumptions; The success probability for a short span is equal to success probability for a long period of time. The success probability in a duration equals to zero as the duration becomes smaller.
The graph of poisson distribution is shown below; Poisson distribution Poisson distribution has the following characteristics; The events are independent of each other, i. An event could occur any number of times in a defined period of time. The average rate of events to take place is constant. Exponential Distribution Like the poisson distribution, exponential distribution has the time element; it gives the probability of a time duration before an event takes place. The graph of exponential distribution is shown below; Exponential distribution The exponential distribution has following characteristics; As shown in the graph, the higher the rate, the faster the curve drops, and lower the rate, flatter the curve.
Also read: Importance of Statistics in Data Science Multinomial Distribution The multinomial distribution is used to measure the outcomes of experiments that have two or more variables. The graph of exponential distribution is shown below; Multinomial Distribution The following are properties of multinomial distribution; An experiment can have a repeated number of trials, for example, rolling of a dice multiple times.
Each trial is independent of each other. The graph of beta distribution is shown below; Beta Distribution The general formulation of beta distribution is also known as the beta distribution of first kind and beta distribution of second kind is another name of beta prime distribution. Referred blog: Conditional Probability Beta-binomial distribution A data distribution is said to be beta-binomial if the Probability of success, p, is greater than zero.
The graph of t-distribution distribution is shown below; T-distribution T-distribution has the following properties; Similar to normal distribution, the t-distribution has bell-shaped curve distribution and is symmetric when mean is zero. The variance is always more than one. Must check: T-test vs Z-test Uniform distribution Uniform distribution can either be discrete or continuous where each event is equally likely to occur.
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