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#### skatenerd

##### Active member

- Oct 3, 2012

- 114

I am given a function:

$$f(x)=\frac{1}{10}(10-x)^2\,;\,0\leq{x}\leq{10}$$

and

$$f(x)=0$$

for all other \(x\).

I need to find the average value of \(x\) where

$$\bar{x}=\frac{\int_{-\infty}^{\infty}x\,f(x)\,dx}{\int_{-\infty}^{\infty}f(x)\,dx}$$

I am not really even sure where to start with this. If the denominator is equal to zero for every value through the integral except for 0 to 10, would I just be able to change it (just the denominator) to

$$\int_{0}^{10}f(x)\,dx$$

I don't think the same would apply for the numerator, because the function is multiplied by \(x\), making it a different function, which is possibly still integratable from \(-\infty\) to \(\infty\).

But wait, there's more. Part (b) is asking to suppose that the variable \(x\) is discrete rather than continuous. Assume \(\Delta{x}=1\) so that \(x\) takes on only integral values 0, 1, 2, ..., 10. Then I need to compute \(\bar{x}\) again and compare to the first part.

So I think after doing part (a), I could maybe know how to attack part (b) if I knew what a discrete variable was...I've never taken a class that ever dealt with discrete variables so I am kind of in the dark here. Any guidance would be very appreciated.