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 understanding probability density functions pdfs the 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 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 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 computing the pdf and plotting it graphically can involve complex hazard rate calculations that use differential equations or integral calculus in practice graphing calculators or statistical software packages are required to calculate a probability density function you should note that investment returns are rarely if ever normally distributed so graphs will likely never be a clean normal distribution curve image by julie bang investopedia 2020example of a probability density functionthe 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 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 the previous three years 