Probability and Computing: Chapter 3 Exercises (cont.2)

Exercise 3.15: Let the random variable X be representable as a sum of random variables X=\sum_{i=1}^{n}X_{i}. Show that, if E[X_{i}X_{j}]=E[X_{i}]E[X_{j}] for every pair of i and j with 1 \leq i <j\leq n, then Var[X] = \sum_{i=1}^{n}Var[X_{i}].

The proof is essentially the same as described in the text. Note that this problem implies we don’t need mutual independence to use linearity of variance, just E[A.B] = E[A]E[B] is enough.

Exercise 3.16: This problem shows that  Markov’s inequality is as tight as it could possible be. Given a positive integer k, describe a random variable X that assumes only nonnegative values such that:

Pr(X \geq kE[X]) = \frac{1}{k} .

Exercise 3.17: Can you give an example(similar to that for Markov’s inequality in Exercise 3.6) that shows that Chebyshev’s in equality is tight? If not, explain why not?

Exercise 3.19: Let Y be a nonnegative integer-valued random variable with postive expectation. Prove

\frac{E[Y]^{2}}{E[Y^{2}]} \leq Pr( Y \ne 0) \leq E[Y].


First we will prove E[Y] \ge Pr(Y \ne 0)

E[Y] = \sum_{y=1}^{\infty}yPr(Y = y) \ge \sum_{y=1}^{\infty} Pr(Y = y) = Pr(Y \ne 0)

Now we prove E[Y]^{2} \le E[Y^{2}]Pr(Y \ne 0)

E[Y]^{2} = \big(\sum_{y=1}^{\infty}yPr(Y=y)\big)^{2}

= \sum_{y=1}^{\infty}\big(yPr(Y=y)\big)^{2} + \sum_{i=1}^{\infty}\sum_{j = i + 1}^{\infty}2ijPr(Y=i)Pr(Y=j)

E[Y^{2}]Pr(Y \ne 0) = \big(\sum_{i=1}^{\infty}i^{2}Pr(Y=i)\big)\big(\sum_{j=1}^{\infty}Pr(Y = j)\big)

= \sum_{y=1}^{\infty}\big(yPr(Y=y)\big)^{2} + \sum_{i=1}^{\infty}\sum_{j=i+1}^{\infty}(i^{2}+j^{2})Pr(Y=i)Pr(Y=j)

By comparing \sum_{i=1}^{\infty}\sum_{j = i + 1}^{\infty}2ijPr(Y = i)Pr(Y = j)
\le \sum_{i=1}^{\infty}\sum_{j=i+1}^{\infty}(i^{2} + j^{2})Pr(Y=i)Pr(Y = j),

we obtain E[Y]^{2} \le E[Y^{2}]Pr(Y \ne 0).

Thus \frac{E[Y]^{2}}{E[Y^{2}]}\le Pr(Y \ne 0). \qquad \square


One response to this post.

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