Definition of Monotonic Probability
Definition
Let’s define a family of probability mass functions or probability density functions for a parameter $\theta \in \mathbb{R}$ and a univariate random variable $T$ as $G := \left\{ g ( t | \theta) : \theta \in \Theta \right\}$. If for all $\theta_{2} > \theta_{1}$, $$ {{ g \left( t | \theta_{2} \right) } \over { g \left( t | \theta_{1} \right) }} $$ is a monotonic function in $\left\{ t : g \left( t | \theta_{1} \right) > 0 \lor g \left( t | \theta_{2} \right) > 0 \right\}$, then $G$ has a Monotone Likelihood Ratio (MLR).
Description
Many widely known distributions, such as the normal distribution, Poisson distribution, binomial distribution, and the exponential family of distributions, can easily be shown to have the Monotone Likelihood Ratio.
See Also
Karlin-Rubin Theorem
If a distribution has a Monotone Likelihood Ratio, the existence of the most powerful test can easily be assured.