**Instructions:** Use Markov's Inequality calculator to estimate an upper bound of the probability of an event \(\Pr(X \ge a)\) according to Markov's Inequality. Please provide the required data in the form below:

## Markov's Inequality calculator

The Markov's Inequality states that for a value \(a > 0\), we have for any random variable \(X\) that takes no negative values, the following upper bound is always observed:

\[\Pr(X \ge a) \le \displaystyle \frac{E(X)}{a} \]Markov's inequality is very important to estimate probabilities, considering its generality in the sense that it applies to any non-negative random variable \(X\).

Indeed, Markov's inequality is crucial to prove a widely used inequality, which is Chebyshev's inequality, and it's the foundation to an even sharper inequality, which is Hoeffding’s Inequality.

### Markov's Inequality Intuition

What is the intuition behind Markov's Inequality? Well, first, there is the clear factor that the probability on a right tail has an upper bound that decreases more and more as we get a more further right tail, which is actually pretty obvious.

Observe the nature of the inequality, which is that \(\frac{E(X)}{a}\) is not the exact value of the probability of the tail, but it is only an upper bound. How close is this bound? Well, now we know that it depends on the actual distribution, but yet there are sharper inequalities like Hoeffding’s Inequality.

But yet, there is a very clear rule in Math: The more general (less specific) the assumptions, the weaker the theorem. So, it is pretty awesome that Markov's inequality exists considering the very general nature of its assumptions.

For example, the empirical rule is a much tighter inequality, but it makes a much stronger assumption: that the underlying distribution is normal. Markov's inequality works for any distribution (of non-negative variable)

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