LFD Book Forum Problem 1.9
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#1
09-29-2015, 01:14 AM
 kongweihan Junior Member Join Date: Sep 2015 Posts: 1
Problem 1.9

I'm working through this problem and stuck on (b).

Since , we get

We also know

Both terms in the desired inequality is bigger than the common term, so I don't know how these two inequalities can lead to the desired conclusion, what did I miss?

Also, in (c), why do we want to minimize with respect to s and use that in (d)?
#2
03-21-2016, 07:29 AM
 MaciekLeks Member Join Date: Jan 2016 Location: Katowice, Upper Silesia, Poland Posts: 17
Re: Problem 1.9

Quote:
 Originally Posted by kongweihan I'm working through this problem and stuck on (b). Since , we get We also know Both terms in the desired inequality is bigger than the common term, so I don't know how these two inequalities can lead to the desired conclusion, what did I miss? Also, in (c), why do we want to minimize with respect to s and use that in (d)?

How do you know that ? I think that is a problem in your proof that you assumed that the joint probability works with Problem 1.9(b) inequality.

To proof (b) I went this way:

1. I used Markov Inequality

2. Problem 1.9(a) gave me this: , hence

Using this the rest of the proof is quite nice to carry out.
#3
05-12-2016, 02:18 AM
 waleed Junior Member Join Date: May 2016 Posts: 5
Re: Problem 1.9

Quote:
 Originally Posted by MaciekLeks How do you know that ? I think that is a problem in your proof that you assumed that the joint probability works with Problem 1.9(b) inequality. To proof (b) I went this way: 1. I used Markov Inequality 2. Problem 1.9(a) gave me this: , hence Using this the rest of the proof is quite nice to carry out.
I don't think the condition right
#4
09-17-2016, 10:43 AM
 svend Junior Member Join Date: Sep 2016 Posts: 2
Re: Problem 1.9

Here's my take on Problem 1.9, part(b), which is following the same lines as the description of MaciekLeks above.

We have:

Since is monotonically increasing in t.

Also, is non negative for all t, implying Markov inequality holds:

The last line being true since [math]x_n[\math] are independent.

From there it directly follows that

#5
11-09-2018, 06:02 PM
 Ulyssesyang Junior Member Join Date: Nov 2018 Posts: 3
Re: Problem 1.9

Quote:
 Originally Posted by svend Here's my take on Problem 1.9, part(b), which is following the same lines as the description of MaciekLeks above. We have: Since is monotonically increasing in t. Also, is non negative for all t, implying Markov inequality holds: The last line being true since [math]x_n[\math] are independent. From there it directly follows that
I just think you should note that there are two expectations, one is based on e^su_n, while other is based on u_n. Of course, you can refer the law of unconscious statisticians to prove that both are same.
#6
03-03-2017, 07:53 AM
 k_sze Member Join Date: Dec 2016 Posts: 12
Re: Problem 1.9

Quote:
 Originally Posted by kongweihan Also, in (c), why do we want to minimize with respect to s and use that in (d)?
I also have trouble understanding (c).

Actually I don't even know how to tackle it. I think I'll need a lot of hand-holding through this one because my math got really rusty since I left school (I'm 34).
#7
03-03-2017, 08:25 AM
 k_sze Member Join Date: Dec 2016 Posts: 12
Re: Problem 1.9

Quote:
 Originally Posted by k_sze I also have trouble understanding (c). Actually I don't even know how to tackle it. I think I'll need a lot of hand-holding through this one because my math got really rusty since I left school (I'm 34).
Will I need to summon notions such as "moment generating function" for part (c) of this problem?
#8
03-15-2017, 06:11 AM
 k_sze Member Join Date: Dec 2016 Posts: 12
Re: Problem 1.9

Quote:
 Originally Posted by k_sze Will I need to summon notions such as "moment generating function" for part (c) of this problem?
So I did end up using moment generating function. And I think the answer to (c) is , using calculus.

But now I'm stuck at (d).

Directly substituting is probably wrong? Because can be simplified to the point where no logarithm appears (unless I made a really big mistake).
#9
03-16-2017, 09:36 AM
 k_sze Member Join Date: Dec 2016 Posts: 12
Re: Problem 1.9

Quote:
 Originally Posted by k_sze So I did end up using moment generating function. And I think the answer to (c) is , using calculus. But now I'm stuck at (d). Directly substituting is probably wrong? Because can be simplified to the point where no logarithm appears (unless I made a really big mistake).
*sigh*

I did end up getting by substituting for , but only after simplifying all the way down, until there is no more or , otherwise I get two powers of 2 with no obvious way to combine them.

So now the remaining hurdle is to prove that .

Yay
#10
04-01-2017, 07:19 AM
 k_sze Member Join Date: Dec 2016 Posts: 12
Re: Problem 1.9

Quote:
 Originally Posted by k_sze *sigh* I did end up getting by substituting for , but only after simplifying all the way down, until there is no more or , otherwise I get two powers of 2 with no obvious way to combine them. So now the remaining hurdle is to prove that . Yay
I finally worked out the proof for . Apparently using Jensen's inequality is supposed to be the simple way, but I simply don't get Jensen's inequality, so I used some brute force calculus (first and second derivatives, find the minimum, etc.)

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