# Markov Inequality The **Markov Inequality** is defined as: $P(X \geq x) \leq \frac{E[X]}{x}$ This is very easy to see visually if we make use of the relationship between the survival function and [Expected Value](Expected%20Value.md), particularly their areas: %206.png) ### Notes on the construction It is worth highlighting a few key notes on the construction since this inequality shows up all over the place: 1. We start of by building a [Probability Model](Probability%20Model.md) that contains a sample space $\Omega$ and a probability measure (maps each $\omega$ to a real number, with constraint that entire measure must sum to 1). 2. We then define a [Random Variable](Random%20Variable.md) $X$ on this model. It maps $\omegas from our sample space to points in $\mathbb{R}$. 3. From here we can enumerate all $\omega \in \Omega$, noting each $P(\omega)$ and $X(\omega)$, and their product (an area). We can use this information to construct a plot of the [Survival-Function](Survival-Function.md), from which we know the entire area of the survival function is equal to $E[X]$. 4. By simply reading off of the graph, we can see that a particular $x$ times the probability of observing a value greater than $x$ is less than the entire area under the curve. 5. However, what allows us to prove this *generally* is that the survival function has two key properties: 1. The survival function is *monotonically decreasing*. If this were not the case then we could end up with the following scenario: %207.png) 2. The survival function is nonnegative (it will never dip below 0). The above constraints are direct consequences of the [Axioms of Probability](Axioms%20of%20Probability) (the first two axioms, i.e. total probability of all outcomes must be 1, and the probability of every event is greater than or equal to 0). These axioms directly ensure the above constraints hold. In a very real sense these constraints provide a structure that we can "grab on to". We can build up logical chains of reasoning, anchoring in to these constraints, making use of them to form complex arguments that must follow from our axioms. ### Strength This is a relatively weak inequality. It is making use of/exploiting very little information (just that pertaining to the expected value). If we think about the true distribution of outcomes, there is far more information present than simply the average value. Yet, the Markov Inequality makes use of the none of that, meaning that the conclusion it draws cannot be *that* strong. --- Date: 20211006 Links to: [Probability MOC](Probability%20MOC.md) Tags: #todo References: * []()