prove Chebyshev's inequality
Understand the Problem
The question is asking to provide a proof for Chebyshev's inequality, which is a statistical theorem that provides an estimate of the probability that a random variable deviates from its mean. It typically involves concepts from probability and statistics.
Answer
P(|X - μ| ≥ kσ) ≤ 1/k².
The final answer is the course of steps proving Chebyshev's inequality: P(|X - μ| ≥ kσ) ≤ 1/k².
Answer for screen readers
The final answer is the course of steps proving Chebyshev's inequality: P(|X - μ| ≥ kσ) ≤ 1/k².
More Information
Chebyshev's inequality is useful in probability theory to bound the probability that a random variable deviates significantly from its mean. One common use is in the proof of the weak law of large numbers.
Tips
A common mistake is not properly determining the correct random variable to which Markov's inequality must be applied. Remember, (X - μ)² is non-negative which allows the use of Markov's inequality.
Sources
- Chebyshev's inequality - Wikipedia - en.wikipedia.org
- Markov and Chebyshev Inequalities - Probability Course - probabilitycourse.com
- Using Markov inequality to prove Chebyshev inequality - math.stackexchange.com
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