Cover information theory 7.21 tall, fat people

Question

I am stuck on Thomas Cover information theory 2nd edition, problem 7.21 Fat, tall people. The problem is like following:

7.21 Tall, fat people. Suppose that the average height of people in a room is 5 feet. Suppose that the average weight is 100 lb.

(a) Argue that no more than one-third of the population is 15 feet tall.

(b) Find an upper bound on the fraction of 300-lb 10-footers in the room.e

For (a), I think height should follow a Gaussian, so if the std is known, then it should not be hard to know 15 feet falls out of how many std away, thus its probability to occur. However, the std is not know?

For (b), I think it is a joint probability of height and weight (iid), so it should be a 2D Gaussian. So asking for upper bound sounds like estimating the std for both?

And this chapter (chapter 7 in "Elements of Information Theory") is particularly about channel capacity and joint typical sequence, so should I use joint typical sequence to solve this problem? But I don't see how is it related. I am really getting stuck, anyone can please give any pointers? Or anyone has a solution I can look at it? Thanks!

(b) P(w≥300, h≥10)=P(w≥300)P(h≥10)≤(5/10)(100/300)=(1/2)*(1/3)=1/6 ?

Answer 1

Don't think about a Gaussian, but of Markov's inequality: $P(X\geq c)\leq\mu_X/c$, so for (a) the answer is $P(X\geq15)\leq5/15$. Can you go from here to (b)?

Answer 2

For (b) I think it's not plausible to believe that people's height and weight are independent random variables. Is a person who's 4 feet tall just as likely to weigh 400 lb as one who is 7 feet tall?

Since you have no control over the correlation I think you'll have to figure things out separately. What is the possible fraction of 300 lb people? What is the possible fraction of 10 footers? Does that help you answer (b)?

Edit: in response to your comments below, I also don't think that independence is the situation in which you have the fewest tall fat people. If 5% (say) of the people are tall couldn't all of them be fat?