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Basics of Expectation and Variance

  1. If the random variable \(X\) takes values in non-negative integers, prove that:

    \(E[X] = \sum_{t=0}^\infty \Pr(X > t)\)

  2. Prove that if \(X_1\) and \(X_2\) are independent random variables then \(E[X_1 \cdot X_2] = E[X_1] E[X_2]\). Is the converse true ?

  3. Let \(X\) be a random variable taking integral nonnegative values. Let \(E[X]\) be its expectation, and let \(E[X^2]\) denote the expectation of its square. Prove that

    \( \Pr(X>0) \geq \frac{E[X]^2}{E[X^2]}\)

    Hint : Use Cauchy-Schwarz inequality.

Source: folklore

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