# Probability Distribution

## What is a 'Probability Distribution'

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 between the minimum and maximum statistically possible values, but where the possible value is likely to be plotted on the probability distribution depends on a number of factors. These factors include the distribution's mean, standard deviation, skewness and kurtosis.

## BREAKING DOWN 'Probability Distribution'

Academics and fund managers alike may determine a particular stock's probability distribution to determine the possible returns that the stock may yield in the future. The stock's history of returns, which can be measured on any time interval, will likely be comprised 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.

## 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. 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. The usual example would use a fair coin and figuring the probability of that coin coming up heads in ten straight flips.

The most commonly used distribution is the normal distribution and it 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.

## Probability Distributions Used in Investing

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. 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 timeframe for a portfolio. Alternatively, an investor can get a probability of loss for an amount of loss and time frame using VaR. Misuse and over-reliance on VaR has been implicated as one of the major causes of the Financial Crisis.