**Exercise 3.15: **Let the random variable be representable as a sum of random variables . Show that, if for every pair of and with , then .

**Solution:**

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 is enough.

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

.

**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 be a nonnegative integer-valued random variable with postive expectation. Prove

.

**Solution:**

First we will prove

Now we prove

By comparing

,

we obtain .

Thus

Posted by http://www.goharshahi.us/member/41460/ on March 15, 2013 at 12:45 pm

I’m gone to say to my little brother, that he should also go to see this blog on regular basis to obtain updated from most up-to-date reports.