Chebyshev's Inequality (probability)

20 Jul, 2023

When we have random variable X and we know it's finite expected value $μ$ and a non-zero variance $σ^{2}$ , we can find a lower bound probability that a random variable falls within some range.

P (μ - t < X < μ + t) \geq 1 - \frac{σ^{2}}{t^{2}}, t > 0

So if we want to know what is the probability that some value falls between for example two standard deviations from the expected value, we can use $t = 2 σ$ and get this:

P (μ - 2 σ < X < μ + 2 σ) \geq 1 - \frac{1}{2^{2}}

which equals $0, 75$ so there's 75% probability of that happening.