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Let $f(x) = \frac{1}{L}$ be a probability function, where $L$ is constant. Find the mean and variance. Discuss your results by making a connection to the moment of inertia definition.

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closed as off-topic by John Rennie, Emilio Pisanty, akhmeteli, Waffle's Crazy Peanut, Qmechanic Oct 21 '13 at 0:36

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2 Answers

You find the mean and variance of probability distributions by computing certain moments of the distribution. The $n^{th}$ moment of a continuous distribution is simply an expectation value of $x^n$ (for the case of f(x) being your distribution). If your distribution f(x) is defined over the interval $[a,b]$, then the mean is given by the first moment:

\begin{equation} \langle x \rangle = \int_a^b dx\ x\ f(x) . \end{equation}

And the variance is given by the second moment minus the square of the first moment:

\begin{equation} \mathrm{Var} = \langle x^2 \rangle - \langle x \rangle^2 \end{equation}

where

\begin{equation} \langle x^2 \rangle = \int_a^b dx\ x^2\ f(x) . \end{equation}

The variance is analogous to the MOI in classical mechanics. See this Wiki article near the bottom for the second part of your question.

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You have to show us your work and where you are stuck. We just won't solve the question for you. Nonetheless, I will offer you a hint:

The mean of a function f(x) is xf(x) (integrated over infinity). Do you know how you could find the variance given the probability function?? Do you understand the meaning of the terms 'mean' and 'variance'? If you do, you should have no problem relating it to the definition for the moment of inertia...

As I said previously, we do not know whether you are well versed in these concepts or if you have even heard of them. Therefore, don't just post a question and expect us to assume everything and solve it. Show your work and give details.

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