What is a Random Variable?
When the value of a
variable
is determined by a chance event, that variable is called a
random variable.
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Discrete vs. Continuous Random Variables
Random variables can be
discrete or
continuous.
- Continuous. Continuous variables, in
contrast, can take on any value within a range of values.
For example, suppose we randomly select an individual from a population.
Then, we measure the age of that person. In theory, his/her age
can take on any value between zero and plus infinity, so age is a continuous
variable. In this example, the age of the person selected is determined by a chance event;
so, in this example, age is a continuous random variable.
Discrete Variables: Finite vs. Infinite
Some references state that continuous variables can
take on an infinite number of values, but discrete variables cannot. This
is incorrect.
- In other cases, however, discrete variables can take on
an infinite number of values. For example, the number of coin flips
that result in heads could be infinitely large.
When comparing discrete and continuous variables, it is more correct to say
that continuous variables can always take on an infinite number of values;
whereas some discrete variables can take on an infinite number of values, but
others cannot.
Test Your Understanding
Problem 1
Which of the following is a discrete random variable?
I. The average height of a randomly selected group of boys.
II. The annual number of sweepstakes winners from New York City.
III. The number of presidential elections in the 20th century.
(A) I only
(B) II only
(C) III only
(D) I and II
(E) II and III
Solution
The correct answer is B.
The annual number of sweepstakes winners results from a random process,
but it can only be a whole number - not
a fraction;
so it is a discrete random variable. The average height of a randomly-selected group
of boys could take on any value between the height of the
smallest and tallest boys, so it is not a discrete variable.
And the number of presidential elections in the 20th century
does not result from a random process;
so it is not a random variable.