This question has been regraded.

A circular dial is randomly spun, and its pointer lands somewhere between 0° and 360°, with every angle equally likely. Let 𝑋 be the angle in degrees where the dial lands. Is the sentence below a true/false statement about X?

X follows a Uniform Distribution, but if we measure the angle in radians, X will cease to follow a Uniform Distribution.

Question

This question has been regraded.

A circular dial is randomly spun, and its pointer lands somewhere between 0° and 360°, with every angle equally likely. Let 𝑋 be the angle in degrees where the dial lands. Is the sentence below a true/false statement about X?

X follows a Uniform Distribution, but if we measure the angle in radians, X will cease to follow a Uniform Distribution.

BlackTom AI · Accepted Answer

Here we evaluate the given statement about X, the angle where the dial lands, which is chosen uniformly from 0° to 360°.
First, we note that when the angle is measured in degrees, X is uniformly distributed on the interval [0°, 360°), since every angle in that range is equally likely.
Second, if we convert the angle to radians by multiplying by π/180, we obtain Y = X · (π/180). This is a linear, one-to-one transformation of a uniform variable on [0, 360°) to a new interval in radians, namely [0, 2π).
Third, a one-to-one linear transformation of a continuous uniform variable preserves the uniformity of the distribution, just on the updated interval. In other words, Y is uniformly distributed on [0, 2π).
Therefore, the statement that X follows a Uniform Distribution in degrees but ceases to be uniform when expressed in radians is incorrect. The distribution remains uniform after the unit change because the transformation is bijective and linear, mapping the entire interval onto a new interval of the same relative structure.
In contrast, some common misconceptions might suggest that changing units alters the fundamental distribution shape; however, in this case it does not, since a uniform distribution is preserved under linear, monotonic scaling of the variable.
So, the assertion as a whole is false, because the uniformity is preserved under the conversion to radians.

ECON 3640-090 Summer 2025 Assignment 7: Random Variables

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