University of Alberta Guide/STAT/222/Formulas and Functions/Probability Density Functions

The probability density function, $$f_{X}(x)\,$$, represents the distribution of the random variable $$X\,$$. Wikipedia defines the PDF as a smoothed out version of the random variable's histogram of its distribution. Basically, the larger the value of $$f_{X}(x)\,$$ for a specific value $$x\,$$ means the more often that value $$x\,$$ will appear when choosing a random variable of its type of distribution.

The total area under the curve formed by $$f_{X}(x)\,$$ will always be one, since that total area represents the probability that the random variable $$X\,$$ will be a value.

PDF for an Exponential RV



One way to calculate the PDF is to differentiate its CDF, $$\frac{\delta}{\delta x}F_{X}(x) = f_{X}(x)\,$$