Inequality for Bernoulli random variable

Inequality for Bernoulli random variable

by Andreea-Alexandra Musat -
Number of replies: 3

Hello, 


In the solution for problem 4 from homework 1, I've seen a reference to an inequality for a Bernoulli random variable (see screenshot attached below). However, as far as I know, I don't have any such inequality in my notes, so could you please clarify which one it is?


Thank you! 



bernoulli rv ineq

In reply to Andreea-Alexandra Musat

Re: Inequality for Bernoulli random variable

by Thomas Weinberger -
Hi Andreea,

We start of on the left hand side with the moment generating function of a Ber(p) random variable and then simply apply Lemma 2.4 from the lecture notes with b=1, a=0 (also check Def. 2.1 for the definition of the sub-gaussian norm if needed).

Best,
Thomas
In reply to Thomas Weinberger

Re: Inequality for Bernoulli random variable

by Andreea-Alexandra Musat -
Hi Thomas,

Thanks for the reply!
However, we cannot use Lemma 2.4 because this lemma is exactly what we're trying to prove in this exercise.

Best,
Andreea
In reply to Andreea-Alexandra Musat

Re: Inequality for Bernoulli random variable

by Thomas Weinberger -
Oops, I just looked at your screenshot without checking the actual problem on the sheet...

I also did not find a suitable inequality in the lecture script, so thanks for pointing that out! 

In this step of the derivation we can apply Hoeffding's lemma. The linked wiki article shows the step-by-step derivation involving the arithmetic mean-geometric mean inequality + Taylor's Theorem.

Hope this helps.

Best,
Thomas