Let L = _n ∈fty _k=0^n e^-n n^kk! Which of the following is the value of L?

GATE DA 2026 Q45: Limit of a Poisson CDF at its mean

Let

$L = \lim_{n \to \infty} \sum_{k=0}^{n} \frac{e^{-n} n^k}{k!}$

Which of the following is the value of $L$ ?

Try it: multiple choice: exactly one correct option

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Answer: A (official GATE 2026 final answer key)

Concept used: The sum is $P(N \le n)$ for $N \sim Poisson(n)$ ; by the central limit theorem $N$ is approximately $Normal(n, n)$ .

First step: $P(N \le n) = P\left(\dfrac{N - n}{\sqrt{n}} \le 0\right) \to \Phi(0)$ .

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Source: official GATE 2026 DA question paper and final answer key, organised by IIT Guwahati. Question paper · Answer key