If X0 has the Poisson distribution π(α), then limα→∞ P{(Xα-α)/(√(α)) ≤u}=Φ(u) for every u. [Hint

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If X0 has the Poisson distribution π(α), then limα→∞...

If $X_{0}$ has the Poisson distribution $\pi(\alpha)$, then $$ \lim _{\alpha \rightarrow \infty} P\left\{\frac{X_{\alpha}-\alpha}{\sqrt{\alpha}} \leq u\right\}=\Phi(u) $$ for every $u$. [Hint: use the Laplace transform $E\left(e^{-\lambda\left(X_{\alpha}-\alpha\right) / \sqrt{\alpha}}\right)$, show that as $\alpha \rightarrow \infty$ it converges to $e^{\lambda^{2} / 2}$, and invoke the analogue of Theorem 9 of $\S$ 7.5.]

If $X_{0}$ has the Poisson distribution $\pi(\alpha)$, then
$$
\lim _{\alpha \rightarrow \infty} P\left\{\frac{X_{\alpha}-\alpha}{\sqrt{\alpha}} \leq u\right\}=\Phi(u)
$$
for every $u$. [Hint: use the Laplace transform $E\left(e^{-\lambda\left(X_{\alpha}-\alpha\right) / \sqrt{\alpha}}\right)$, show that as $\alpha \rightarrow \infty$ it converges to $e^{\lambda^{2} / 2}$, and invoke the analogue of Theorem 9 of $\S$ 7.5.]

Key Concepts

Convergence in Distribution

Convergence in distribution is a concept in probability theory where a sequence of random variables converges to a limiting random variable in terms of their cumulative distribution functions. In this context, it means that the normalized Poisson variable, as its parameter tends to infinity, converges to the standard normal distribution. This type of convergence is crucial in the application of the CLT for describing asymptotic behavior.

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental result in probability theory which states that, under certain conditions, the sum (or appropriately normalized sum) of a large number of independent and identically distributed random variables tends toward a normal distribution, regardless of the original distribution of the variables. This theorem underpins the result showing that the normalized Poisson variable converges to a normal (Gaussian) distribution as the parameter increases.

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, under the assumption that these events occur with a constant mean rate and independently of the time since the last event. It is commonly parameterized by its mean (often denoted by alpha), and it is particularly useful for modeling count data and rare events.

Laplace Transform

The Laplace transform is an integral transform used to convert a function of a real variable into a function of a complex variable. In probability theory, the Laplace transform of a distribution serves a role similar to that of moment generating functions, as it can be used to characterize the distribution, study convergence properties, and simplify the analysis of sums of random variables.

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