A probability distribution is a statistical function that outlines all possible values and likelihoods that a random variable can achieve within a defined range. This range is bounded by minimum and maximum values, yet the precise likelihood of any outcome depends on factors such as mean (average), standard deviation, skewness, and kurtosis.
Key Insights
- A probability distribution represents the expected outcomes of possible values in a given data-generating process.
- They come in various forms, each characterized by its mean, standard deviation, skewness, and kurtosis.
- Investors utilize probability distributions to forecast asset returns and mitigate risks.
How Probability Distributions Operate
One of the most common forms of probability distributions is the normal distribution, often referred to as the “bell curve.” Several different distributions are prevalent in various applications. The phenomena underlying a data-generating process typically determine its probability distribution, known as the probability density function.
Probability distributions form the basis for cumulative distribution functions (CDFs), which provide cumulative probabilities and range from 0% to 100%.
Academics, analysts, and fund managers may determine a stock’s probability distribution to shed light on potential future returns, relying on historical data to lower sampling error by using larger sample sizes.
Varieties of Probability Distributions
There are diverse types of probability distributions, such as normal distribution, chi-square distribution, binomial distribution, and Poisson distribution, each serving different purposes.
Binomial Distribution
The binomial distribution assesses the probability of an event happening multiple times over a set number of trials with a specific probability in each trial. For instance, it could measure the likelihood of a basketball player hitting a number of free throws in a game. A fair coin toss in successive trials can also illustrate a binomial distribution, which is discrete since it yields only binary outcomes (1 or 0).
Normal Distribution
A normal distribution is widely used across finance, investing, science, and engineering. It relies on mean and standard deviation for characterization, depicted as a symmetric bell-shaped curve. This distribution, defined by a mean of zero and standard deviation of 1.0, sees approximately 68% of data within one standard deviation from the mean, 95% within two, and 99.7% within three. Unlike binomial distribution, it’s continuous, encapsulating all possible values.
Application of Probability Distributions in Investing
Although stock returns might be assumed to follow a normal distribution, they often exhibit higher kurtosis with more frequent extreme returns than a normal distribution predicts. Stock returns are sometimes described as log-normal, reflecting a distribution with rich tails.
Investors leverage probability distributions in risk management to appraise potential portfolio losses based on historical return distributions. A significant risk management tool is value-at-risk (VaR), which estimates the minimum potential loss within a specified timeframe and probability.
Illustrative Example: Rolling Dice
Consider the possible sums when rolling two six-sided dice. Each die has a 1/6 chance of showing any number from 1 to 6. However, the distribution of sums forms a different probability distribution: seven emerges as the most common sum (formed by pairs like 1+6 and 6+1), while sums of two and twelve are the rarest.
Validity of Probability Distributions
To ensure a probability distribution’s validity, follow these steps:
- Check if each probability exceeds or equals zero and does not exceed 1.
- Ensure the sum of all probabilities equals 1.
Financial Applications of Probability Distributions
In finance, probability distributions are used to forecast investment returns and evaluate loss probabilities. This allows investors to strategize risk mitigation.
Common Probability Distributions
Some of the most familiar probability distributions include uniform, binomial, Bernoulli, normal, Poisson, and exponential distributions.
Conclusion
Probability distributions elucidate all possible outcomes for a random variable, playing a crucial role in investment decision-making and risk management. Understanding these distributions helps in plotting stocks’ potential, ultimately enhancing strategic financial planning.
Related Terms: mean, standard deviation, skewness, kurtosis, sampling error, value-at-risk (VaR).
References
- U.S. Securities and Exchange Commission. “Remarks Before the Peterson Institute of International Economics”.
