mode

Given a probability distribution (density) function $f_X(x)$ with random variable $X$ and $x\in ℝ$, a mode of $f_X(x)$ is a real number $\alpha$ such that:

1. $f_X(\alpha )\ne \min (f_X(x))$,

2. $f_X(\alpha )\ge f_X(z)$ for all $z\in ℝ$.

The mode of $f_X$ is the set of all modes of $f_X$ (It is also customary to say denote the mode of $f_X$ to be elements within the mode of $f_X$). If the mode contains one element, then we say that $f_X$ is unimodal. If it has two elements, then $f_X$ is called bimodal. When $f_X$ has more than two modes, it is called multimodal.

• if $\mathrm{\Omega }=\{0,1,2,2,3,4,4,4,5,5,6,7,8\}$ is the sample space for the random variable $X$, then the mode of the distribution function $f_X$ is 4.

• if $\mathrm{\Omega }=\{0,2,4,5,6,6,7,9,11,11,14,18\}$ is the sample space for $X$, then the modes of $f_X$ are 6 and 11 and $f_X$ is bimodal.

• For a binomial distribution with mean $np$ and variance $np(1-p)$, the mode is

  $$\{\alpha \mid p(n+1)-1\le \alpha \le p(n+1)\}.$$

• For a Poisson distribution with integral sample space and mean $\lambda$, if $\lambda$ is non-integral, then the mode is the largest integer less than or equal to $\lambda$; if $\lambda$ is an integer, then both $\lambda$ and $\lambda -1$ are modes.

• For a normal distribution with mean $\mu$ and standard deviation $\sigma$, the mode is $\mu$.

• For a gamma distribution with the shape parameter $\gamma$, location parameter $\mu$, and scale parameter $\beta$, the mode is $\gamma -1$ if $\gamma >1$.

• Both the Pareto and the exponential distributions have mode = 0.