What Is a Probability Density Function (PDF)?

The probability density function (PDF) is an expression used in statistics 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 commonly used by 澳洲幸运5官方开奖结果体彩网:financial analysts to understand how returns are꧅ distri🌺buted to evaluate the risk and expectations of investment prices and returns.

Understanding Probability Dꦫensity Functions (PDFs)

Statistical tools are commonly used by people in the investment world. They are very useful to analyze and determine market trends along with understanding the potential 澳洲幸运5官方开奖结果体彩网:risks and returns associated with certain investments. This allows investors and financial pr💎ofessionals to make more well-informedꦗ decisions about how to invest their money.

The probability density function is a statistical measurement of how often investment returns fall within a specified range. PDFs are typically depicted on a graph, with a normal 澳洲幸运5官方开奖结果体彩网:bell curve indicating neutral market risk, and a skewed curve at either end indicating gr🐭eater or lesser risk-reward.

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

The image below demonstrates 澳洲幸运5官方开奖结果体彩网: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).

Computing the PDF and plotting it graphically can involve 澳洲幸运5官方开奖结果体彩网:complex hazard rate calculations that use differential equations or integral calculus. In practice, graphing calculators or statistical software packages are requir🐟ed to calculate a proꦡbability density function.

Example of a Probability Density Function (PDF)

The probability density function measures continuous variables. Having said that, it's important to note that stock and investment returns are generally not continuous random variables. Rather, they are discrete. However, most financial analysts assume that returns and prices are continuous so they can model performance and analyze risks.

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

The Bottom Line

澳洲幸♐运5官方开奖结果体彩网:Probability distri༺bution functions (PDFs) describe the expected values of random variables drawn from a sample. The shape of the PDF explains how likely it is ꦛthat an observed value might occur.

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