Probability Density Function

Probability Density Function

The probability density function of a continuous random variable is represented by its probability density function (or density function). It is a function f_X(x) whose graph satisfies the following conditions:

1. The area bounded by the function curve (or line) and the x-axis is equal to 1.
2. The probability that the random variable assumes a value within the interval (a,b) is equal to the area bounded by the curve (or line)of f_X(x), the x-axis, and the two ordinates x = a and
   x = b.

 

Using this definition of a probability density function, the probability that a continuous random variable assumes a specific value will be zero. This is true because the probability that a random variable takes on an exact value a is the probability that the random variable assumes a value in the interval (a, a); in the density function's graph, the area bounded by the curve, the x-axis, and two parallel lines that both pass through (b,0), is zero.

Intuitively, this fact about continuous random variables states that a continuous random variable does not assume a specific value, but rather it takes on a value in a specified interval with a certain probability. For example, the random variable Z whose value is the height of a person in California is a continuous random variable. The probability that Z has the value 66 (inches) is zero, but the probability that the value of Z is between 64 and 68 is some positive value.

NOTE: For a continuous random variable X, its probability density function f_X(x) satisfies the following equality:

Since most probability density functions have a non-linear graph, the evaluation of the areas under the curve of these density functions requires knowledge of differentiation and integration, two basic calculus concepts. Since most basic statistics courses and books do not assume a knowledge of calculus, the treatment of continuous probability functions is often left to more advanced courses and texts. This CD-ROM statistics reference will not cover continuous probability distributions in depth; future editions will assume a basic knowledge of calculus and will thus be able to address this subject with greater depth.

EX. The probability density function of the continuous random variable X is as follows:

To show that f_X(x) is indeed a density, we must show that the area bounded by the line graph of f_X(x) and the x-axis is equal to 1. The area in question is that of a right triangle, with legs of length 1 and 2. Thus, its area is 0.5(1)(2) = 1.

To find the value of P(0.5 < X < 1.5), we must find the area bounded by the line graph of f_X(x), the x-axis, and the ordinates x = 0.5 and x = 1.5. This area is that of a trapezoid, with parallel sides of length 0.25 and 0.75, and a base of length 1. Thus, its area is 0.5(1)(0.25 + 0.75) = 0.5.