# Moment of Inertia [closed]

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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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:

$$\langle x \rangle = \int_a^b dx\ x\ f(x) .$$

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

$$\mathrm{Var} = \langle x^2 \rangle - \langle x \rangle^2$$

where

$$\langle x^2 \rangle = \int_a^b dx\ x^2\ f(x) .$$

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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