Calculate $P(\bar{X} < \epsilon)$ given that for all $i=1,2,...,n$, $P(X_i < \epsilon) > 1 - a$, $X$ is taken from some distribution $D$ and $0<a<1$

Question

My goal is to find whether we could end up with something in the form $P(\bar{X} < \epsilon) > 1 - b^n$ where $n$ is the amount of data points we have

Answer 1

Sorry but you did not provide enough condition to get you a definite answer for that. We need the full probability distribution function in order to calculate that.

To illustrate my point, here are an examples that both satisfy your condition but gives you different answers to your question.

$$ P_1(x) = \left\{ \begin{aligned} 1-a \quad & \text{when}\quad x=0 \\ a \quad & \text{when}\quad x=2\epsilon \\ \end{aligned} \right. $$ $$ P_2(x) = \left\{ \begin{aligned} 1-a \quad & \text{when}\quad x=\epsilon/2 \\ a \quad & \text{when}\quad x=2\epsilon \\ \end{aligned} \right. $$