We fix $p, q \in [0,1]$ and $(X_i)_{1 \leq i \leq n}$ a family of $n$ random variables taking values in $\{0,1\}$ mutually independent Bernoulli random variables with parameter $p$. We set $S_n = \sum_{i=1}^n X_i$. We assume in this question that $p < q$. a. Justify that $$\mathbb{P}\left(\left|\frac{S_n}{n} - q\right| \leq \left|\frac{S_n}{n} - p\right|\right) = \mathbb{P}\left(S_n \geq \frac{p+q}{2}n\right)$$ b. Let $X$ be a Bernoulli random variable with parameter $p$. For $u > 0$, calculate $\mathbb{E}\left(e^{uX}\right)$. c. Show that for all $u > 0$, $$\mathbb{P}\left(S_n \geq \frac{p+q}{2}n\right) \leq e^{-n\left(\frac{p+q}{2}u - \ln\left(1-p+pe^u\right)\right)}$$ Hint. One may assume that if $(Z_i)_{1 \leq i \leq n}$ are $n$ mutually independent random variables taking a finite number of values, then $\mathbb{E}\left(\prod_{i=1}^n Z_i\right) = \prod_{i=1}^n \mathbb{E}\left(Z_i\right)$. d. Show that $\mathbb{P}\left(S_n \geq \frac{p+q}{2}n\right) \leq e^{-n\frac{(p-q)^2}{2}}$.
We fix $p, q \in [0,1]$ and $(X_i)_{1 \leq i \leq n}$ a family of $n$ random variables taking values in $\{0,1\}$ mutually independent Bernoulli random variables with parameter $p$. We set $S_n = \sum_{i=1}^n X_i$. We assume in this question that $p < q$.
a. Justify that
$$\mathbb{P}\left(\left|\frac{S_n}{n} - q\right| \leq \left|\frac{S_n}{n} - p\right|\right) = \mathbb{P}\left(S_n \geq \frac{p+q}{2}n\right)$$
b. Let $X$ be a Bernoulli random variable with parameter $p$. For $u > 0$, calculate $\mathbb{E}\left(e^{uX}\right)$.
c. Show that for all $u > 0$,
$$\mathbb{P}\left(S_n \geq \frac{p+q}{2}n\right) \leq e^{-n\left(\frac{p+q}{2}u - \ln\left(1-p+pe^u\right)\right)}$$
Hint. One may assume that if $(Z_i)_{1 \leq i \leq n}$ are $n$ mutually independent random variables taking a finite number of values, then $\mathbb{E}\left(\prod_{i=1}^n Z_i\right) = \prod_{i=1}^n \mathbb{E}\left(Z_i\right)$.
d. Show that $\mathbb{P}\left(S_n \geq \frac{p+q}{2}n\right) \leq e^{-n\frac{(p-q)^2}{2}}$.