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

1. $f_X(\alpha)\neq \operatorname{min}(f_X(x))$ ,
2. $f_X(\alpha)\geq f_X(z)$ for all $z\in \mathbb{R}$ .

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 $\Omega=\lbrace 0,1,2,2,3,4,4,4,5,5,6,7,8 \rbrace$ is the sample space for the random variable $X$ , then the mode of the distribution function $f_X$ is 4.
• if $\Omega=\lbrace 0,2,4,5,6,6,7,9,11,11,14,18 \rbrace$ 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 $$\lbrace \alpha \mid p(n+1)-1\leq \alpha \leq p(n+1) \rbrace.$$
• 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.