Which of the following is a property of a probability density function ($pdf$)?

• A. The $pdf$ is only defined for discrete random variables.
• B. The $pdf$ must always be greater than $1$.
• C. The total area under the curve of the $pdf$ is equal to $1$.
• D. The $pdf$ can take negative values for some values of $x$.

Answer: (C)

Verified step by step guidance

1. Understand that a probability density function (pdf) is used to describe continuous random variables, not discrete ones. So, the pdf is not defined for discrete random variables; instead, probability mass functions (pmf) are used for discrete cases.
2. Recall that a pdf must be non-negative for all values of the random variable \(x\). This means the pdf cannot take negative values anywhere, as probabilities cannot be negative.
3. Recognize that the value of a pdf at any point \(x\) can be greater than 1, especially if the distribution is very concentrated, so the statement that the pdf must always be greater than 1 is incorrect.
4. Focus on the fundamental property of a pdf: the total area under the curve of the pdf over the entire range of \(x\) must equal 1. This represents the fact that the total probability of all possible outcomes is 1.
5. Summarize that the correct property of a pdf is: \(\text{the total area under the curve of the pdf is equal to } 1\).