One of the problems of my Statistics assignment requires me to calculate the expected value of a minimum. Here's the situation:
The following density function is given: $f(x) = \frac{\theta}{x^2}$ where $x\ge\theta$ and $\theta>0$
I have to calculate: E[min{$X_i$}]
My initial guess was that the smallest possible value of $X$ is $\theta$ sense $x\ge\theta$ so the expected value of the minimum would be $\theta$, but then again, in an acquired sample, you can't be 100% certain that the smallest possible value of $X$ will be one of the observations. So what then?