Table of Contents
- What Is a Probability Distribution?
- Key Takeaways
- How Probability Distributions Work
- Important Note on Stock Returns
- Discrete Probability Distribution vs. Continuous Probability Distribution
- Types of Probability Distributions
- Binomial Distribution
- Normal Distribution
- Poisson Distribution
- Probability Distributions Used in Investing
- Probability Distribution and the Central Limit Theorem
- Example of a Probability Distribution
- What Makes a Probability Distribution Valid?
- How Are Probability Distributions Used in Finance?
- What Are the Most Commonly Used Probability Distributions?
- What Is the Difference Between Probability and Odds?
- What Is the Law of Large Numbers?
- The Bottom Line
What Is a Probability Distribution?
Let me tell you directly: a probability distribution in statistics shows you the relative likelihood of each possible outcome within a specific timeframe. As a stock analyst, I use it to plot potential movements in a stock's price based on its historical data.
You'll often see it displayed as a bell curve for quick reference, and it can help project things like the minimum and maximum resources a business might need next year or the chances of a severe flu outbreak versus milder ones.
Essentially, probability distribution is a mathematical branch focused on calculating the likelihood of every possible event within a set of variables.
Key Takeaways
Here's what you need to grasp: a probability distribution reveals the most and least probable occurrences based on variables. Investors rely on it to project asset returns over time and hedge against losses. The two main types are discrete and continuous probability distributions.
How Probability Distributions Work
The most common one you'll encounter is the normal distribution, or bell curve. The underlying process of a phenomenon dictates its distribution, which we call the probability density function.
You can use these to create cumulative distribution functions (CDFs) that sum up probabilities cumulatively, starting at zero and ending at 100%.
In my work, whether as an academic or financial analyst, I determine a stock’s probability distribution to evaluate possible future returns.
Important Note on Stock Returns
Remember, a stock’s return history is measured from recent intervals to avoid sampling error, which you can reduce by increasing your sample size.
Discrete Probability Distribution vs. Continuous Probability Distribution
You need to understand these two fundamentals: discrete distributions handle countable outcomes, like the number of heads in coin flips or customers in a store per hour. They're choppy because outcomes are distinct.
Continuous distributions, on the other hand, cover variables with infinite values in a range, like heights or task times, leading to smoother curves.
Types of Probability Distributions
There are many types, including normal, chi-square, binomial, and Poisson, each for different data processes.
Binomial Distribution
This one evaluates the probability of an event over multiple trials, like free throws in basketball (1 for basket, 0 for miss) or heads in coin flips. It's discrete, with only binary outcomes.
Normal Distribution
The normal distribution is the go-to in finance, science, and engineering, defined by mean and standard deviation. It's symmetric, bell-shaped, with no skew and kurtosis of 3. About 68% of data falls within one standard deviation, 95% within two, and 99.7% within three. Unlike binomial, it's continuous.
Poisson Distribution
This discrete distribution models independent events in a fixed interval, like customer arrivals or emails received, where events are rare and the rate is constant.
Probability Distributions Used in Investing
Stock returns show kurtosis with fat tails, often lognormal due to prices bounded at zero but unlimited upside. In risk management, distributions evaluate portfolio losses via metrics like value at risk (VaR), which gives minimum loss probability over time—though overreliance on VaR contributed to the 2008 crisis.
Probability Distribution and the Central Limit Theorem
The central limit theorem states that sums of many independent variables approach a normal distribution with large samples, allowing inferences about populations. For example, average heights in samples will form a bell curve.
Example of a Probability Distribution
Consider rolling two dice: each has a 1/6 chance per number, but sums peak at seven (most ways to get it) and drop to two and twelve (least likely).
What Makes a Probability Distribution Valid?
It's valid if each probability is between zero and one, and they all sum to one.
How Are Probability Distributions Used in Finance?
They estimate investment returns and hedge against potential losses.
What Are the Most Commonly Used Probability Distributions?
Uniform, binomial, Bernoulli, normal, Poisson, and exponential.
What Is the Difference Between Probability and Odds?
Probability is favorable outcomes over total; odds are probability of happening over not happening, like 1:3 for 0.25 probability.
What Is the Law of Large Numbers?
As trials increase, results approach the true probability, making sample means converge to population means for reliable inference.
The Bottom Line
Probability distributions list all possible values for a random variable, crucial in investing for stock performance and loss evaluation.
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