Estimate using Chebyshev´s inequality

Task number: 3552

Assume that we throw a fair die one hundred times. Let the random variable \(X\) equal the sum of all values thrown.

Using Chebyshev's inequality estimate the probability \(P[|X-350|\ge 50]\).

  • Hint

    Decompose \(X\) into a sum of many independent random variables.

    Chebyshev's inequality is \(P[|X-EX|\ge t] \le \frac{var(X)}{t^2}\).

  • Solution

    If the random variable \(Y_i\) is the value of the \(i\)-th throw of the die, then \(X=Y_1+Y_2+…+Y_{100}\), and all hundred variables are mutually independent.

    For each variable \(Y_i\) we have \(EY_i=\frac72\), \(var(Y_i)=\frac{35}{12}\).

    \(EX=E[Y_1+Y_2+…+Y_{100}]=100\cdot \frac72=350\).

    \(var(X)=\sum_{i=1}^{100} var(Y_i)=\frac{3500}{12}\).

    Now by Chebyshev's inequality

    \(P[|X-350|\ge 50] \le \frac{3500}{12 {\cdot} 50^2}=\frac{7}{60}\doteq 0{,}117\)

  • Answer

    The given probability is less than 12 %.

Difficulty level: Moderate task
Routine calculation training
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