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The question is: The uniform density function given in the notes has median and mean $= 3.5$, by inspection. Calculate the variance. I have the answer key, but I don't understand it. (http://pages.stat.wisc.edu/~ifischer/Intro_Stat/Lecture_Notes/Lec_Notes_Sols/4.4.pdf problem $12$).

I have two questions. First, how does the domain change from $(-\infty, \infty)$ to $(1, 6)$? How do I know $f(x) = \frac15$? Thank you!

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The original note is here. Look for example $1$.

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We are given that the corresponding distribution is the continuous uniform distribution with support being $(1,6)$.

The density function would be $f(x) = \frac{1}{6-1}\mathbb{1}_{(1,6)}$. Which is why we can integrate over the support as $f(x)=0$ outside the support.

Remark: Given the mean and median alone and the information that the distribution is uniform is not sufficient to determine the corresponding support.

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