# The Basics of Probability Density Function (PDF), With an Example

## What Is a Probability Density Function (PDF)?

The probability density function (PDF) is a statistical expression that defines the probability that some outcome will occur. In this function, the probability is the percentage of a dataset's distribution that falls between two criteria.

PDF is used by financial analysts to understand how returns are distributed in order to evaluate the risk and expectations of investment prices and returns.

### Key Takeaways

• Probability density functions are a statistical measure used to gauge the likelihood that an investment will have returns that fall within a range of values and indicate the risks involved.
• PDFs are usually plotted on a graph that typically resembles a bell curve, with the data lying below the curve.
• A skewed curve at either end indicates greater or lesser risk/reward.

## Understanding Probability Density Functions (PDFs)

Probability density function is a measurement of how often investment returns fall within a specified range. PDFs are typically depicted on a graph, with a normal bell curve indicating neutral market risk, and a skewed curve at either end indicating greater or lesser risk-reward.

Skewness is a shift of the taller portion of the curve to the right or left. If the curve is shifted to the left with a long tail on the right (right skew), analysts consider it to suggest there is greater upside reward. If it is shifted to the right with a long tail to the left (left skew), analysts suggest that there is greater downside risk.

The image below demonstrates normally distributed data with a bell curve. The data mean is the line in the middle, and the vertical lines are standard deviations, or how far data falls from the mean.

The first two vertical lines on either side of the mean show that 68.5% of the data fall within +/-1 standard deviation from the mean. So, if this were a curve of normally distributed stock returns, you would see that 68.5% of the time, returns fall between the -1 SD and +1 SD lines and that market risk is neutral (there is no skew).

You should note that investment returns are rarely, if ever, normally distributed, so graphs will likely never be a clean normal distribution curve.

## Calculating PDF

Computing the PDF and plotting it graphically can involve complex calculations that use differential equations or integral calculus. In practice, graphing calculators or statistical software packages are required to calculate a probability density function.

## Example of a Probability Density Function

The probability density function measures continuous variables—stock and investment returns are generally not continuous random variables; they are discrete. However, most financial analysts assume that returns and prices are continuous so that they can model performance and analyze risks.

In the image below, the S&P 500 index values from May 20, 2020, to May 19, 2023, were sequenced and plotted. The result was a bell curve with right skew, indicating the possibility of greater upside reward over the previous three years.

## What Does a Probability Density Function (PDF) Tell Us?

A probability density function (PDF) describes how likely it is to observe some outcome resulting from a data-generating process. A PDF can tell us which values are most likely to appear vs. the less likely outcomes. This will change depending on the shape and characteristics of the PDF.

## What Is the Central Limit Theorem (CLT) and How Does It Relate to PDFs?

The central limit theorem (CLT) states that the distribution of a random variable in a sample will begin to approach a normal distribution as the sample size becomes larger, regardless of the true shape of the distribution. Thus, we know that flipping a coin is a binary process, described by the binomial distribution (heads or tails).

However, if we consider several coin tosses, the odds of getting any particular combination of heads and tails begin to differ. For instance, if we were to flip the coin ten times, the odds of getting 5 of each is most likely, but getting ten heads in a row is extremely rare. Imagine 1,000 coin flips, and the distribution approaches the normal bell curve.

## What Is a PDF vs. a CDF?

A probability density function (PDF) explains which values are likely to appear in a data-generating process at any given time or for any given draw. A cumulative distribution function (CDF) instead depicts how these marginal probabilities add up, ultimately reaching 100% (or 1.0) of possible outcomes. Using a CDF, we can see how likely it is that a variable's outcome will be less than or equal to some predicted value.

## The Bottom Line

Probability distribution functions (PDFs) describe the expected values of random variables drawn from a sample. The shape of the PDF explains how likely it is that an observed value might occur.

Stock prices and returns tend to follow a log-normal distribution rather than a normal one, indicating that downside losses are more frequent than very large gains, relative to what the normal distribution would predict.

Article Sources
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1. Wall Street Journal. "S&P 500 Index."

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