What is the expected value in a non-normal distribution?

Question

I'm really confused about an elementary aspect of distributions, and I can't quite seem to find an answer to this in my old stats texts.

I'm having a hard time understanding how the expected value or mean of a continuous distribution could be different from the median or the point where the probability dist function integrates to .5. And yet non-normal distributions like the log normal dist have different means and medians:

I think my whole intuition about the expected value was that it was the point where the pdf evaluates to .5. Can someone explain expected values more clearly than my stats text does?

Answer 1

The mean $\mu$ and the median $\tilde\mu$ are equal if they exist and the pdf $f$ is symmetric with respect to $x=\mu$, that is, if $f(\mu-x) =f(\mu+x)$ for every $x>0$.

Answer 2

Maybe it is easier to see with a finite distribution. Suppose the values are $1,1,1,2,2,2,2,2,5,5,100$. The median is $2$ because half the values are above and half below. The mean is $\frac 1{11}(1+1+1+2+2+2+2+2+5+5+100)\approx 12.2$. The mode is again $2$ because there are more of them than anything else.

Which is useful depends on what you are using it for. The mean will give the average value over many samples. It gets pulled most by one outlier, here the $100$. The median might be "what you are close to" most of the time. The mode is the most likely value of a single sample.