When is a random variable considered uniformly distributed?

A random variable is considered uniformly distributed when each outcome within a specified range has an equal chance of occurring. This characteristic of uniform distribution can be clearly seen in the case where the probability is constant across the range of outcomes.

While the choice noted as correct points to proportionality related to the interval’s length, this is a feature that can also apply to continuous uniform distributions. In a continuous uniform distribution, the probability density function is flat, meaning that for any equal-length subintervals within the range, the probabilities are proportional to the lengths of those intervals.

This means that if you were to divide the entire interval into smaller segments, each segment would have a probability equal to its length divided by the total length of the interval, ensuring that the total probability integrates to 1 over the entire range.

In contrast, the remaining choices describe situations that either do not maintain equal probabilities across outcomes, such as decreasing probabilities over time, or suggest probabilities that do not represent a uniform distribution. The choice referring to zero probability would indicate that an outcome could never occur, which is fundamentally at odds with the concept of uniform distribution where each outcome must be possible and equally likely.