Statistics Dictionary
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Probability Density Function
Most often, the equation used to describe a
continuous probability distribution
is called a
probability density
function
. Sometimes, it is referred to as a density function,
a PDF, or a pdf. For a continuous probability
distribution, the density function has the following properties:
-
Since the continuous random variable is defined over a continuous range of
values (called the domain
of the variable), the graph of the density function will also be continuous
over that range.
-
The area bounded by the curve of the density function and the x-axis is equal
to 1, when computed over the domain of the variable.
-
The probability that a random variable assumes a value between a and b
is equal to the area under the density function bounded by a and b.
For example, consider the probability density function shown in the graph below.
Suppose we wanted to know the probability that the random variable X was
less than or equal to a. The probability that X is less than or
equal to a is equal to the area under the curve bounded by a and
minus infinity - as indicated by the shaded area.
Note: The shaded area in the graph represents the probability that the random
variable X is less than or equal to a. This is a
cumulative probability
. However, the probability that X is exactly
equal to a would be zero. A continuous random variable can take on an
infinite number of values. The probability that it will equal a specific value
(such as a) is always zero.