Given that $X_n$ is an exponential random variable with parameter $λ=n$.
How does $P(X_n≥ε)=e^{-nε}$ ?
According to this probability course, the equality holds since $X_n∼Exponential(n)$.
Honestly, that's no real explanation; it's only common sense and already provided by the context.
Is there a formula for this? I can't seem to find anything online.
What I do know is that based on the WLLN:
$$P(X_n≥ε)=P\big(\big|\frac{X_1+...+X_n}{n}\big|≥ε\big)$$
So, by substituting in $X_n∼Exponential(n)$, this is what I get
$$P(X_n≥ε)=P\big(\big|\frac{e^1+...+e^n}{n}\big|≥ε\big)$$
Am I on the right track? How does this simplify to $e^{-nε}$?
I would highly appreciate any input on this. Thanks!