Which statement about probability distributions is incorrect?

Question

BlackTom AI · Accepted Answer

The question asks which statement about probability distributions is incorrect, but the provided data for options is incomplete.
Option present: 'a. Probability densities are between 0 and 1.'
Evaluation of this statement: In probability theory, a probability density function (for a continuous random variable) is nonnegative for all x and its integral over the entire space equals 1. However, the value of a density at a single point can exceed 1 for some distributions (especially when the distribution is highly concentrated over a small interval). Therefore, the blanket claim that densities are between 0 and 1 is not universally true, making this statement incorrect in general.
Why this is a misconception: Some students think densities must always lie in [0,1], conflating the density value with probabilities. But probabilities are integrals of the density over intervals, not the point value of the density itself. A density can be greater than 1 as long as its area under the curve remains 1.
Why this is incorrect in context: Since the defining properties of a density are nonnegativity and integrals to 1, the restriction to [0,1] for the density values is unnecessarily strict and false in general.
About the missing options: The provided data shows no other answer options to compare. If there were additional statements (b, c, d, etc.), each would need its own assessment to determine whether it correctly describes probability distributions or contains a misconception. Given only option a is available, the analysis focuses on why option a is not universally true and thus is the incorrect statement in a typical set where other options would be correct.
In summary, the key point is that probability densities are nonnegative and integrate to 1, but their point values can exceed 1; hence the statement that they are between 0 and 1 is not generally correct.

FIT5201 - MUM S1 2025 Quiz 1 2025 S1

Which statement about probability distributions is incorrect?

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