- #1

longrob

- 52

- 0

From my book. We have,

[tex]p(x)=\begin{cases}

\frac{1}{2} & -2\leq x\leq-1\,\textrm{or}\;1\leq x\leq2,\\

0 & \text{otherwise~}.\end{cases}[/tex]

So, the kth moment is given by

[tex]M_{k}=\frac{1}{2}\int_{-2}^{-1}x^{k}dx+\frac{1}{2}\int_{1}^{2}x^{k}dx[/tex]

So, obviously,

[tex]M_{k}=(1+(-1)^{k})\frac{1}{2}\int_{1}^{2}x^{k}dx[/tex]

Is this a correct explanation of this step:

Since p(x) is an even function, it follows that the left integral is exactly equal to the right integral when k is even, and exactly equal to -1 times the right integral when k is odd. Can it be explained better/ more rigorously ? Will this explanation always apply to any even real-valued function of x ?