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\label{PROBL-EXP-CHERNOFF} (‘naive exponential estimate’) Let [math]Y_1,\dots,Y_d[/math] be independent random variables and assume that [math]|\E(Y_i^k)|\leqslant k![/math] holds for all [math]k\geqslant0[/math] and [math]i=1,\dots,d[/math].

• Use the series expansion of [math]\exp(\cdot)[/math] and the assumption to get
  [[math]] \E(\exp(tY_i))\leqslant \Bigsum{k=0}{\infty}t^k=\left\{\begin{array}{cl}\textfrac{1}{1-t} &\text{ for } t\in(-1,1),\\\infty & \text{otherwise.}\end{array}\right. [[/math]]
• Show by means of calculus that
  [[math]] \inf_{t\in(0,1)}\exp(-ta)\prod_{i=1}^d\smallfrac{1}{1-t}=\left\{\begin{array}{cl}(\textfrac{a}{d})^d\exp(d-a) &\text{ if } a \gt d,\\ 1 & \text{otherwise.}\end{array}\right. [[/math]]
• Derive an estimate for [math]\P\bigl[|Y_1+\cdots+Y_d|\geqslant a\bigr][/math] from the above.
• Compare the bound in (iii) with the bound of Theorem.