A probability density function over its entire domain must integrate to what value?

A probability density function over its entire domain must integrate to what value?

• A. 0
• B. 1
• C. The mean
• D. The median

Answer: (B)

The main idea is that a probability density function must account for all possible outcomes, so the total probability across its entire domain is 1. For a continuous random variable, the probability that X falls within any interval is the area under the curve f(x) over that interval, i.e., the integral of f(x) on that interval. When you integrate over the whole domain, you’re summing all possible probabilities, which must equal 1. If the integral were 0 or any number other than 1, the distribution wouldn’t be properly normalized. The mean is a different concept: it’s the expected value, computed as the integral of x f(x) dx, a weighted average of values, not the total area. The median is the value that splits the probability mass into two equal halves, also not the total area.

The main idea is that a probability density function must account for all possible outcomes, so the total probability across its entire domain is 1. For a continuous random variable, the probability that X falls within any interval is the area under the curve f(x) over that interval, i.e., the integral of f(x) on that interval. When you integrate over the whole domain, you’re summing all possible probabilities, which must equal 1. If the integral were 0 or any number other than 1, the distribution wouldn’t be properly normalized.

The mean is a different concept: it’s the expected value, computed as the integral of x f(x) dx, a weighted average of values, not the total area. The median is the value that splits the probability mass into two equal halves, also not the total area.