Markov & Chebyshev bounds
Markov’s inequality says P(X ≥ a) ≤ E[X]/a for a non-negative X — a bound from the mean alone. Chebyshev sharpens it using the variance: P(|X − μ| ≥ kσ) ≤ 1/k². Both are deliberately crude, holding for EVERY distribution with the stated moments, which is exactly what makes them useful when the distribution is unknown.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Check non-negativity for MarkovX ≥ 0 is required. Without it the bound is false.
- Apply MarkovP(X ≥ a) ≤ E[X]/a. Useless when the result exceeds 1, which happens for small a.
- Apply Chebyshev in units of σP(|X − μ| ≥ kσ) ≤ 1/k². For a two-sided bound at distance d, use k = d/σ.
- Expect a loose answerThese bounds are worst-case. A bound of 0.25 against a true probability of 0.05 is normal and still correct.
Worked example
X has mean 50 and standard deviation 5. Bound P(|X − 50| ≥ 15).
- The deviation 15 equals 3 standard deviations, so k = 15/5 = 3.
- Chebyshev: P(|X − μ| ≥ kσ) ≤ 1/k².
- = 1/9.
- This holds whatever the distribution of X.
Answer. P(|X − 50| ≥ 15) ≤ 1/9 ≈ 0.111. For a normal distribution the true value is about 0.0027, showing how loose the bound is.
Where marks get dropped
These are the specific errors that cost credit on markov & chebyshev bounds questions — QED's rubric penalises each of them separately.
- Applying Markov to a variable that can be negative, which makes the inequality invalid.
- Reporting a bound greater than 1 as a probability. It is a valid but vacuous bound — say so.
- Using Chebyshev with the variance where the standard deviation is required. The k in the formula counts SDs.
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Markov & Chebyshev bounds — frequently asked questions
Why are these bounds so weak?
Because they must hold for every distribution with that mean and variance, including the worst possible one. Extra assumptions like normality give far tighter bounds.
How is Chebyshev derived from Markov?
Apply Markov to the non-negative variable (X − μ)², whose expectation is σ². The event |X − μ| ≥ kσ is the same as (X−μ)² ≥ k²σ².
What are they used for?
Proving the weak law of large numbers, and bounding tails in algorithm analysis where the distribution is unknown but the moments are computable.
The rest of Probability
Events, conditional probability, random variables. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Sample spaces & events
- 2Conditional probability & independence
- 3Bayes’ theorem
- 4Random variables & expected value
- 5Variance & standard deviation
- 6Binomial & uniform distributions
- 7Law of total probability & tree diagrams
- 8Geometric & Poisson distributions
- 9Joint, marginal & conditional distributions
- 10Markov & Chebyshev bounds
- 11Equally likely outcomes & counting
- 12Linearity of expectation
- 13Indicator random variables
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