The moment-generating function of the binomial distribution is $$m_X(s)=(pe^s+1-p)^n$$
Let $X$ be the number of heads in $n$ coin tosses. Let $X$ be binmoially distributed with $p=1/2$. How can I use the following inequalities to estimate the probability of at least $80$ heads in $100$ coin tosses?
a) $P(X \ge t)\le \frac{E(X^+)}{t},t>0$
b) $P(X \ge t)\le \inf_{s\ge0} e^{-st}m_X(s)$
Our $t$ is $80$ but what is $E(X^+)$? And for b) how can I determine the infimum of $e^{-st}m_X(s)$ for $s>=0$?
Thanks in advance!
$X^+$ is the non-negative part of X, so in Binomial $X^+=X$ and from (a) we get
$P(X \ge 80)\le \frac{100\cdot 0.5}{80}$
b) $P(X \ge 80)\le \inf_{s\ge0} e^{-80s}(0.5e^s+0.5)^{100}$
and you have to find the Monotonicity of function $f(s)=e^{-80s}(0.5e^s+0.5)^{100}$ , from
$f'(s)=-80e^{-80s}(0.5e^s+0.5)^{100}+50e^{-79s}(0.5e^s+0.5)^{99}$
https://www.wolframalpha.com/input/?i=solve-80e%5E%7B-80s%7D(0.5e%5Es%2B0.5)%5E%7B100%7D%2B50e%5E%7B-79s%7D(0.5e%5Es%2B0.5)%5E%7B99%7D%3D0