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Math Help - Question using Chebyshev's inequality

  1. #1
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    Question using Chebyshev's inequality

    Right, fingers crossed this questions come out OK:
    Question using Chebyshev's inequality-q.png

    My inital thought is where do I start? I don't quite understand the question
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  2. #2
    Senior Member Sambit's Avatar
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    Since X_i's have mean \mu and variance \sigma^2 for each i, then \bar{X} has mean \mu and variance \frac{\sigma^2}{n}. Consequently, the Chebyshev's inequality can be rewritten for \bar{X} as:

    P(|\bar{X}-\mu|\geq a)\leq \frac{\sigma^2}{na^2}

    =>P(\bar{X}\geq \mu+a, \bar{X}\leq \mu-a)\leq\frac{\sigma^2}{na^2} (breaking the modulus sign)

    So \frac{\sigma^2}{na^2} is the required upper limit.
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