easy Probability
A release-checked easy problem for training Exponential/Gamma, Density Transform, Order Statistics.
Let $X_1,X_2,\dots \sim \exponential{\lambda}$ IID. Let $p_n = \prob{M_n \leq \dfrac{\ln(n)}{\lambda}}$, where $M_n = \max(X_1,\dots,X_n)$. Compute $\displaystyle \lim_{n\rightarrow \infty} p_n$ when $\lambda = 4$. The answer is in the form $e^{-a}$ for some $a$. Find $a$.
This Probability problem is tagged Exponential/Gamma, Density Transform, Order Statistics. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.
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