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

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

Answer: (D)

Verified step by step guidance

1. Recall that a probability density function (pdf) describes the relative likelihood of a continuous random variable taking on a particular value.
2. Understand that a pdf is a continuous function, not a discrete one, so it cannot be described as always discrete.
3. Remember that the pdf must be non-negative for all values of \( x \), meaning \( f(x) \geq 0 \) for all \( x \); it cannot take negative values.
4. Note that the value of the pdf at any point \( x \) can be less than or equal to 1; it does not have to be greater than 1 everywhere.
5. The key property of a pdf is that the total area under the curve over the entire range of \( x \) must equal 1, which mathematically is expressed as \( \int_{-\infty}^{\infty} f(x) \, d x = 1 \).