Chernoff Bound. Inequalities like $P(Y \ge 150) \le e^{-50log(\frac{27}{16})} $ . Let $X_1, X_2 , ...,X_{200} $ be independent Bernoulli random variables. 
In other words: $X_i$ ~ $Ber(\frac{1}{2}$) for $ i \in $ {1,...,200}.
Define $ Y = \sum_{i=1}^{200} X_i $. 
Given: $Y$ ~ $Bin(200,\frac{1}{2}) $.
Show that:
a) $ P(Y \ge 150) \le \frac{2}{3} $
b) for every $ t$ $\in$ $\mathbb{R}$ :
$\mathbb{E}(e^{t Y}) $ = $ e^{200log(\frac{1}{2}e^t+\frac{1}{2})} $
c) for every $t$ $\in \mathbb{R}$ : $P(Y \ge 150) \le e^{-150t+200log(\frac{1}{2}e^t+\frac{1}{2})} $
d) Find the optimal $t$ and show that: $P(Y \ge 150) \le e^{-50log(\frac{27}{16})}  $
My attempt:
a) Solved this with Markov's inequality. 
b) $\mathbb{E}(e^{t Y}) = \mathbb{E}(e^{t \sum_{i=1}^{200} X_i }) = \mathbb{E}(e^{ \sum_{i=1}^{200} t  X_i }) = \mathbb{E}( \prod_{i=1}^{200}e^{t  X_i }) $ . I know that $X_i$ is independent. But how do I know that $ e^{tX_i}$ is independent? Anyways now I'm using the independence: $ \prod_{i=1}^{200} \mathbb{E}(e^{t  X_i }).$ Now I need that: $\mathbb{E}( e^{t  X_i }) = \frac{1}{2}e^t+\frac{1}{2}.$ Can someone explain me why this equality holds? The rest of b) is clear.
c) I was able to show this for $t > 0 $: $P(Y \ge 150)= P(e^{tY} > e^{t150})$. Use Markov's inequality: So $P(e^{tY} > e^{t150}) < \frac{1}{e^{t150}} \mathbb{E}(e^{t Y}) = e^{-150t+200log(\frac{1}{2}e^t+\frac{1}{2})}.$ What can I do for $t \le 0 $?
d) Unfortunately I have no idea for this part. 
I hope you can help me. :) Edit: c) is clear now because I have to show 
this inequality only for $ t \ge 0 $ 
 A: b) The independence comes from a standard theorem that if $X_1, X_2, \dots, X_n$ are independent, then $f_1(X_1), f_2(x_2), \dots, f_n(X_n)$ are independent for any functions $f_i$. For the expectations, try the Law of the Unconscious Statistician:
$$\mathbb E[e^{t X_i}] = e^{t \cdot 1} \cdot \frac 1 2 + e^{t \cdot 0} \cdot \frac 1 2.$$
c) Show that when $t < 0$, the exponent on $e$ is positive, hence the bound is trivial.
d) Hint: Recall from early calculus how to find optimal values of functions; the upshot is always to take a derivative, set it to 0, and solve.
Editing to expand this hint: Suppose you wanted to optimize the function $f(x) = -3x^2 + 5x + 7$. In this case, "optimize" would mean "maximize," although in general it could refer to a max or a min depending on the context. For differentiable functions, maxima and minima can occur only where there are horizontal tangents. Here, that would mean setting $0 = f'(x) = -6x + 5$ to obtain $x = 5/6$, the $x$-coordinate of the maximum value.
In your problem, your task is to optimize the function $f(x) = e^{-150t + 200 \log(\frac 1 2 e^t + \frac 1 2 )}$. Since the function $e^x$ is an increasing function, it attains a maximum (or minimum) where its exponent does. Thus, your task is actually to optimize the function $g(x) = -150t + 200 \log(\frac 1 2 e^t + \frac 1 2)$. Can you take it from here?
