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.

类似问题

Which of the following characterizes a continuous random variable for which all outcomes between some minimum and maximum value are equally likely?

Which of the following characterizes a continuous random variable for which all outcomes between some minimum and maximum value are equally likely?

Name the Supreme Court decision that removed financial constraints on PACs.

Political scientists Joshua Kalla and David Broockman find that senior policy makers made themselves available to _____ between three and four times more often than they did to constituents.

Name the political scientist who argues that parties are designed to serve the needs of ambitious politicians (namely candidates and officeholders).

Name the political scientist who said this famous quote about the problems of pluralism: “The flaw in the pluralist heaven is that the heavenly choir sings with a strong upper-class accent.”

What is the name of the third party candidate in 2000 who was not allowed to participate in the presidential debates? Hint: We watched a video clip with him in class.

E.E. Schattshneider said, "Democracy is ________ save in terms of the parties."

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单