Probability Density Function
The probability density function, f(x), describes the behavior of a random variable. Typically, the probability density function is viewed as the shape of the distribution. Consider the histogram of the length of fish shown in the figure below.
The probability density function is similar to the overlaid model in the figure above. The area below the probability density function to the left of a given value, x, is equal to the probability of the random variable represented on the x-axis being less than the given value x. Since the probability density function represents the entire sample space, the area under the probability density function must equal one. Since negative probabilities are impossible, the probability density function, f(x), must be positive for all values of x. Stating these two requirements mathematically,
and f(x)> 0 for continuous distributions. For discrete distributions
for all values of n, and f(x)> 0.
The area below the smooth curve in the figure above is greater than one, thus this curve is not a valid probability density function. The density function representing the data in this figure is shown below.
The figure below demonstrates how the probability density function is used to compute probabilities. The area of the shaded region represents the probability of a single fish, drawn randomly from the population having a length less than 185. This probability is 15.9%.
The figure below demonstrates the probability of the length of one randomly selected fish having a length greater than 220.
The area of the shaded region in the figure below demonstrates the probability of the length of one randomly selected fish having a length greater than 205 and less than 215. Note that as the width of the interval decreases, the area, and thus the probability of the length falling in the interval decreases. This also implies that the probability of the length of one randomly selected fish having a length exactly equal to a specific value is zero. This is because the area of a line is zero.
Example: A probability density function is defined as
f(x) = a/x , 1<x<10
For f(x) to be a valid probability density function, what is the value of a?
Solution: To be a valid probability density function, all values of f(x) must be positive, and the area beneath f(x) must equal one. The first condition is met by restricting a and x to positive numbers. To meet the second condition, the integral of f(x) from one to ten must equal 1.