Discrete versus Continuous Random Variables
A random variable may be discrete or continuous.
- Discrete Random Variable
- Continuous Random Variable
Discrete Random Variable:
If X can take only a countable number of distinct values such as 0,1,2……, then X is called a discrete RV. Note that discrete RVs are usually counts (but not necessarily).
Note: If a random variable takes only a finite number of distinct values, then it must be discrete.
Example: A coin is tossed 10 times. Let X = number of heads that are noted. Here X can only take values 0,1,2,3…..10. So, X is a discrete random variable. We can list all possible values of a discrete RV.
Example: Any number of coin flips. Note that outcomes will always be integer values. You will never have HALF or QUARTER heads or tails.
Continuous Random Variable:
Continuous is opposite to discrete. Continuous random variable takes an infinite number of possible values. We cannot list or count all possible values of continuous RV. Suppose listing all possible values between 0 and 1 is not possible, because there are infinite number of values between 0 and 1. It can take any value in a range (or interval).
A continuous RV is not defined at specific values. Instead it is defined over an interval of values and is represented by the area under a curve (i.e., integration).
Let X is a continuous RV. Suppose P(X=0.234) = probability of RV X is equal to 0.234. Understand that P(X=0.234) = 0, because we can’t assign probability to a certain value.
Note that number of telephone calls arriving at an office in a finite time is an example of discrete random variable. Exact time of arrival of a call is an example of continuous random variable. Height of a person can take any positive real value. So height is a continuous random variable.
