A continuous random variable is not defined at specific values. Instead, it is defined over an interval of values, and is represented by the area under a curve (in advanced mathematics, this is known as an integral). The probability of observing any single value is equal to 0, since the number of values which may be assumed by the random variable is infinite.[1]

 

Unlike in discrete random variables, we can no longer give the probability of each value because it is impossible to say what each of these precise values is. We focus on a particular level of accuracy and the probability of getting a range of values.

 

For discrete probability distributions, we look at the probability of getting a particular value; for continuous probability distributions, we look at the probability of getting a particular range.

 

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.[2]

 

The characteristics of continuous random variables are:[3]

• The outcomes are measured, not counted.
• Geometrically, the probability of an outcome is equal to an area under the density curve, f (x).
• Each individual value has zero probability of occurring. Instead we find the probability that the value is between two endpoints. I.e. the probability of X being equal to 4 is zero, however we can find the probability that X is between 5.2 and 7.6.

 

For continuous random variables, the total probability is equal 1 and the total area under the curve is equal to 1. Each actual value has a zero probability of happening so we can’t find the probability that x=4 or x=2 but we can find the probability that x=1.5 or x=

 

For continuous random variables, probabilities are given by area.

 

To find the probability of getting a particular range of values, start off by sketching the probability density function.

 

The probability density function f(x) is a function that you can use to find the probabilities of a continuous variable across a range of values. It tells us what the shape of the probability distribution is.

 

The probability of getting a particular range of values is given by the area under the line between those values.