Context: See "2 Hoeffding’s Inequality" in : http://www.stat.cmu.edu/~larry/=stat705/Lecture2.pdf
My particular question arises within 'section 2 Hoeffding's Inequality' is: $$ e^{-tn\varepsilon }\prod_{i}^{}\mathop{\mathbb{E}}(e^{tY_{i}}) = e^{-tn\varepsilon }(\mathop{\mathbb{E}}(e^{tY_{i}}))^{n}. $$ I did not understand how the these two are equal. I think I understand now, but I want some confirmation...if Y1 = frequency of heads in 3 coin flips, Y2 = frequency of heads in 2 coin flips...
$$Y_{1}\neq Y_{2}$$
BUT, the expection$$\mathop{\mathbb{E}}(Y_{1})=\mathop{\mathbb{E}}(Y_{2})=.5$$ and so,
$$\mathop{\mathbb{E}}(e^{tY_{1}})=\mathop{\mathbb{E}}(e^{tY_{2}})?$$ It's the equal expectations that allows the removal of product notation and use the exponent 'n'. Is this correct?
Yes, your understanding is correct.
It is due to the expectations are equal.$$\mathop{\mathbb{E}}(e^{tY_{1}})=\ldots = \mathop{\mathbb{E}}(e^{tY_{n}})$$
I prefer to remove the $i$ on the right hand side though and replace it with a particular index. $$ e^{-tn\varepsilon }\prod_{i}^{}\mathop{\mathbb{E}}(e^{tY_{i}}) = e^{-tn\varepsilon }(\mathop{\mathbb{E}}(e^{tY_{1}}))^{n}. $$