Summaries of Distributions

Median

• We say that is a median of a random variable $X$ if $P(X\le c)\ge 1/2$ and $P(X\ge c)\ge 1/2$

Mode

• For a discrete r.v. , we say that is a mode of if it maximizes the PMF: $P(X=c)/geqP(X=x)$ for all $x$.

Moments

• The $n$th moment of an r.v. $X$ is $E(X^n)$
  • Let be an r.v. with mean $\mu$ and variance ${\sigma }^2$. For any positive integer : the $n$th moment of $X$ is **$E(X^n)$
  • the $n$th central moment of $X$ is **$E((X-\mu {)}^n)$
  • the $n$th standardized moment of $X$ is $E((\frac{(X-\mu )}{\sigma }{)}^n)$

Skewness

• the skewness of an r.v. with mean $\mu$ and variance ${\sigma }^2$ is the third standardized moment of $X$: Skew(X) = $E((\frac{(X-\mu )}{\sigma }{)}^3)$