how does mean, expected value and variance differ?

Question

Could someone please explain how mean ,expected value and 1st moment differs? I get that the expected value incorporates some kind of probability distribution but why does everyone keep saying the mean = 1st moment??? Can we square the 1st moment to get the 2nd moment? Does not make sense to me how can you square the mean or expected value to get the variance (or 2nd moment)????

Answer 1

Let $F(x)$ be the cumulative distribution function for a random variable $X$. Then the $k^{th}$ moment is DEFINED as $\mu_k=E(X^k)=\int_ {-\infty}^\infty x^kdF(x)$.. If a density function $f(x)$ exists, then the moment can be represented as $\int x^kf(x)dx$. By definition the first moment is the mean, while the variance is $\sigma^2(X)=E(X-\mu_1)^2=\mu_2-\mu_1^2$.