Inequality for Bernoulli random variable

Inequality for Bernoulli random variable

par Andreea-Alexandra Musat,
Nombre de réponses : 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

En réponse à Andreea-Alexandra Musat

Re: Inequality for Bernoulli random variable

par 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
En réponse à Thomas Weinberger

Re: Inequality for Bernoulli random variable

par 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
En réponse à Andreea-Alexandra Musat

Re: Inequality for Bernoulli random variable

par 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