What does the area under the line of a continuous random variable's graph represent?

What does the area under the line of a continuous random variable's graph represent?

• A. It must equal one
• B. It can exceed one
• C. It is always less than the total frequency
• D. It varies based on the variable type

Answer: (A)

The area under the curve of a continuous random variable's probability density function (PDF) represents the total probability of all possible outcomes within a given range. For any continuous random variable, the total area under the curve must equal one. This reflects the certainty that some outcome will occur within that range, as probabilities must sum up to one for any complete sample space. In the context of probability distributions, if the area were to exceed one or be less than one, it would contradict the fundamental principles of probability, wherein the total likelihood of all potential outcomes cannot logically amount to anything other than one. Thus, stating that the area under the line must equal one efficiently encapsulates the idea that we are accounting for all possibilities of the random variable.

The area under the curve of a continuous random variable's probability density function (PDF) represents the total probability of all possible outcomes within a given range. For any continuous random variable, the total area under the curve must equal one. This reflects the certainty that some outcome will occur within that range, as probabilities must sum up to one for any complete sample space.

In the context of probability distributions, if the area were to exceed one or be less than one, it would contradict the fundamental principles of probability, wherein the total likelihood of all potential outcomes cannot logically amount to anything other than one. Thus, stating that the area under the line must equal one efficiently encapsulates the idea that we are accounting for all possibilities of the random variable.