Probability Distributions: Discrete vs. Continuous
All probability distributions can be classified as
discrete probability distributions or as
continuous probability distributions, depending on
whether they define probabilities associated with discrete
variables or continuous variables.
Discrete vs. Continuous Variables
If a variable can take on
any value between two specified values, it is called a continuous variable;
otherwise, it is called a discrete variable.
Some examples will clarify the difference between discrete and continuous
Suppose the fire department mandates that all fire fighters must weigh between
150 and 250 pounds. The weight of a fire fighter would be an example of a
continuous variable; since a fire fighter's weight could take on any value
between 150 and 250 pounds.
Suppose we flip a coin and count the number of heads. The number of heads could
be any integer value between 0 and plus infinity. However, it could not be any
number between 0 and plus infinity. We could not, for example, get 2.5 heads.
Therefore, the number of heads must be a discrete variable.
Just like variables,
probability distributions can be classified as discrete or continuous.
Discrete Probability Distributions
If a random variable
is a discrete variable, its
probability distribution is called a discrete probability
An example will make this clear. Suppose you flip a coin two times. This simple
statistical experiment can have four possible outcomes: HH, HT, TH, and
TT. Now, let the random variable X represent the number of Heads that result
from this experiment. The random variable X can only take on the values 0, 1,
or 2, so it is a discrete random variable.
The probability distribution for this statistical experiment appears below.
|Number of heads
The above table represents a discrete probability distribution because it
relates each value of a discrete random variable with its probability of
occurrence. In subsequent lessons, we will cover the following discrete
Note: With a discrete probability distribution, each possible value of the
discrete random variable can be associated with a non-zero probability. Thus, a
discrete probability distribution can always be presented in tabular form.
Continuous Probability Distributions
If a random variable
is a continuous variable, its
probability distribution is called a continuous probability
A continuous probability distribution differs from a discrete probability
distribution in several ways.
Instead, an equation or formula is used to describe a continuous probability
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:
- 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.
In subsequent lessons, we will cover the following continuous probability