From 1f6d7db006a0fbefea9228490fc5e2a57206ffe6 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Thu, 7 May 2026 11:01:58 +0200 Subject: [PATCH] [PS] Improve central limit theorem --- .../parts/05_limit-theorems/03_central-limit-theorem.tex | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/semester4/ps/ps-jh/parts/05_limit-theorems/03_central-limit-theorem.tex b/semester4/ps/ps-jh/parts/05_limit-theorems/03_central-limit-theorem.tex index 827b931..8115e84 100644 --- a/semester4/ps/ps-jh/parts/05_limit-theorems/03_central-limit-theorem.tex +++ b/semester4/ps/ps-jh/parts/05_limit-theorems/03_central-limit-theorem.tex @@ -15,7 +15,7 @@ $\cX_k$ i.i.d mit $\E[\cX_k] = \mu$, $\V[\cX_k] = \sigma^2$. Für Partialsummen \limit{n}{\8} \P \left[ \frac{S_n - n\mu}{\sigma \sqrt{n}} \leq x \right] = \Phi(x) \] -\shortremark $\E[S_n] = n \mu$, $\V[S_n] = n \sigma^2$; $S_n^* = \frac{S_n - \mu}{\sigma \sqrt{n}} \! \overset{\text{approx}}{\sim} \! \cN(0, 1)$ +\shortremark $\E[S_n] \! = \! n \mu$, $\V[S_n] \! = \! n \sigma^2$; $S_n^* \! = \! \frac{S_n - n\mu}{\sigma \sqrt{n}} \! \overset{\text{approx}}{\sim} \! \cN(0, 1)$ für grosse $n$, mit $\overset{\text{approx}}{\sim}$ gespr. ``approx. gleichverteilt gemäss''. Also ist für $\E[S_n^*] = 0$ und $\V[S_n^*] = 1$. @@ -24,3 +24,5 @@ bzw. $\overline{\cX}_n \! \overset{\text{approx}}{\sim} \! \cN \left( \mu, \frac \shortremark Für $S_n \sim \text{Bin}(n, p)$ ist $S_n \overset{\text{approx}}{\sim} \cN(np, np(1 - p))$ und $\P[a < S_n \leq b] \approx \Phi \left( \frac{b + \frac{1}{2} - np}{\sqrt{np(1 - p)}} \right) - \Phi \left( \frac{a + \frac{1}{2} - np}{\sqrt{np(1 - p)}} \right)$ + +\shortremark Für $\P[S_n \leq y]$: ZGS verwenden mit $\displaystyle x = \frac{y - n\mu}{\sigma \sqrt{n}}$