In the context of continuous probability distributions, the area under the entire probability density curve is equal to which of the following values?

• A. $100$
• B. $0$
• C. $0.5$
• D. $1$

Answer: (D)

Verified step by step guidance

1. Recall the definition of a probability density function (PDF) for a continuous random variable: the total area under the PDF curve over the entire range of possible values must equal 1.
2. This is because the area under the curve represents the total probability, and the probability of all possible outcomes combined must be 1 (or 100%).
3. Mathematically, this is expressed as the integral of the PDF \( f(x) \) over its entire domain \( (-\infty, \infty) \):
4. \[ \int_{-\infty}^{\infty} f(x) \, dx = 1 \]
5. Therefore, the area under the entire probability density curve cannot be 0, 0.5, or 100; it must be exactly 1.
6. This fundamental property ensures that the PDF properly describes a valid probability distribution.