The area under the probability density function between two values represents the probability that a continuous random variable falls within that interval.

The area under the probability density function between two values represents the probability that a continuous random variable falls within that interval.

• A. True
• B. False
• C. It cannot be determined from the information given
• D. Only for discrete variables

Answer: (A)

In a continuous distribution, probabilities come from area under the density curve. The probability that the variable X falls between two values a and b is the integral of the density f(x) from a to b, which geometrically is the area under the curve between a and b. This is why the area between two points represents that probability. Remember that the total area under the curve over its entire support is 1, and the area of a single point is zero, so exact values have probability zero in continuous cases. This area interpretation doesn’t apply to discrete variables, where probabilities are sums of point masses. So the statement is true for continuous random variables.

In a continuous distribution, probabilities come from area under the density curve. The probability that the variable X falls between two values a and b is the integral of the density f(x) from a to b, which geometrically is the area under the curve between a and b. This is why the area between two points represents that probability. Remember that the total area under the curve over its entire support is 1, and the area of a single point is zero, so exact values have probability zero in continuous cases. This area interpretation doesn’t apply to discrete variables, where probabilities are sums of point masses. So the statement is true for continuous random variables.