Which statement describes the conditions for a valid PDF?

Which statement describes the conditions for a valid PDF?

• A. f(x) <= 0 for all x, and the total area under the PDF equals 1
• B. f(x) >= 0 for all x, and the total area under the PDF equals 1
• C. f(x) >= 0 for all x, and the total area under the PDF equals 0
• D. f(x) can be negative but the integral equals 1

Answer: (B)

A valid PDF must assign nonnegative probability to every outcome and must carry total probability mass of 1 across its entire domain. That means f(x) must be greater than or equal to zero for all x, and when you integrate f(x) over the whole domain, the result should be 1. The nonnegativity ensures you’re not giving negative probabilities to any event, and the integral equal to 1 ensures the entire probability space sums to one. If any of these conditions fail—if f(x) is negative somewhere, or if the total area under the curve is 0, or if you only required the integral to be 1 but allowed negative values—then the function wouldn’t describe a valid probability distribution.

A valid PDF must assign nonnegative probability to every outcome and must carry total probability mass of 1 across its entire domain. That means f(x) must be greater than or equal to zero for all x, and when you integrate f(x) over the whole domain, the result should be 1. The nonnegativity ensures you’re not giving negative probabilities to any event, and the integral equal to 1 ensures the entire probability space sums to one. If any of these conditions fail—if f(x) is negative somewhere, or if the total area under the curve is 0, or if you only required the integral to be 1 but allowed negative values—then the function wouldn’t describe a valid probability distribution.