Show that limn →∞ e^-n ∑k=0^n (n^k)/(k !)=(1)/(2) by applying the central limit theorem to suita

step 1 Okay, suppose Xn is a sequence of iid Poisson distribution with parameter 1. We know define Sn t

Show that limn →∞ e^-n ∑k=0^n (n^k)/(k !)=(1)/(2) by...

Show that
$$
\lim _{n \rightarrow \infty} e^{-n} \sum_{k=0}^{n} \frac{n^{k}}{k !}=\frac{1}{2}
$$
by applying the central limit theorem to suitably chosen, independent, Poisson distributed random variables.

Key Concepts

Normal Approximation to the Poisson Distribution

When the parameter of a Poisson distribution is large, the distribution can be approximated by a normal distribution with mean equal to the parameter and variance also equal to the parameter. This approximation is a direct consequence of the CLT and is crucial in evaluating probabilities such as the likelihood of a Poisson random variable being less than or equal to its mean.

Additivity Property of Poisson Random Variables

An important property of Poisson random variables is that the sum of independent Poisson random variables is also Poisson distributed, with the parameter equal to the sum of the individual parameters. This allows us to express a Poisson random variable with a large parameter n as the sum of many independent Poisson variables with smaller parameters, which facilitates the application of the CLT.

Central Limit Theorem

The Central Limit Theorem (CLT) is a key result in probability theory that states that the sum of a large number of independent, identically distributed random variables, when properly normalized, tends toward a normal distribution regardless of the underlying distribution, provided the variance is finite. This is used to approximate the distribution of the sum of Poisson random variables in the limit, allowing us to compute probabilities related to the sum using the normal distribution.

Poisson Distribution

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. It is fundamental for modeling rare events, and in this context, serves as the basis for the sum that is being approximated.

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