## Chebyshev inequality - enigma or simplicity itself?

Let's go back to the very basics. The true probability distribution is usually unknown. This is why using separate values and probabilities is prohibited and we work with various averages. However, as you will see below, the Chebyshev inequality answers a question about behavior of certain probabilities.

### Motivation

Table 1. Income distribution

Income | Percentage | P(Income>=c) | Chebyshev bound | Bound/true |
---|---|---|---|---|

10 | 0.027 | 1 | 5.05 | 5.05 |

20 | 0.066 | 0.973 | 2.525 | 2.595 |

30 | 0.123 | 0.907 | 1.683 | 1.856 |

40 | 0.179 | 0.784 | 1.263 | 1.611 |

50 | 0.202 | 0.606 | 1.01 | 1.667 |

60 | 0.179 | 0.403 | 0.842 | 2.089 |

70 | 0.123 | 0.225 | 0.721 | 3.204 |

80 | 0.066 | 0.102 | 0.631 | 6.186 |

90 | 0.027 | 0.036 | 0.561 | 15.583 |

100 | 0.009 | 0.009 | 0.505 | 56.111 |

I simulated this distribution using the normal distribution function of Excel in such a way as to get relatively low percentages of poor and rich people, see Figure 1.

Suppose the government wants to increase the income tax on wealthy people and use the resulting tax revenue to support the low income population. The question is what part of the population will be impacted. The percentage of the population with income higher than or equal to a given cut-off level

(1)

For example, the government may decide to impose a higher tax on wealthy people with

(2)

The third column of Table 1 contains these probabilities for all cut-off values.

**Question. **The true probabilities are usually unknown but the mean is normally available (to obtain the GDP per capita, just divide the GDP by the head count). In our case the mean income is 50.5. What can be said about (1) if the the cut-off value and the mean are known?

### Chebyshev's answer

Chebyshev noticed that for those

Thus, we cannot find the exact value of (1) but we can give an upper bound

The above proof applies to any nonnegative random variable **simplest form of the Chebyshev inequality**:

(3)

### Extensions

- If
changes sign, its absolute value is nonetheless nonnegative, so . -
It is more interesting to bound the probability of deviation of

from its mean . For this, just plug in (3): . -
One more step allows us to obtain

instead of at the right. Note that the events and are equivalent. Therefore

The result we have obtained **the Chebyshev inequality**.

### Digression

A long time ago I read a joke about P.L. Chebyshev. He traveled to Paris to give a talk named "On the optimal fabric cutout". The best Paris fashion designers gathered to listen to his presentation. They left the room after he said: For simplicity, let us imagine that the human body is ball-shaped.

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