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Steffo 2020-01-29 16:42:11 +01:00
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uni.steffo.eu

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{"/":{"style.5336e.css":{"type":"style","weight":1},"bundle.24f07.js":{"type":"script","weight":1}}}
{"/":{"style.5336e.css":{"type":"style","weight":1},"bundle.a0fa5.js":{"type":"script","weight":1}}}

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@ -728,6 +728,12 @@ export default class Statistica extends Component {
<p>
<Latex>{r`\forall \epsilon > 0, P([ \left| X - E(X) \right| \geq \epsilon]) \leq \frac{Var(X)}{\epsilon^2}`}</Latex>
</p>
<p>
E anche:
</p>
<p>
<Latex>{r`\forall \epsilon > 0, P([ \left| X - E(X) \right| < \epsilon]) \geq 1 - \frac{Var(X)}{\epsilon^2}`}</Latex>
</p>
<Example>
Serve per semplificare i calcoli quando la funzione di ripartizione è difficile da calcolare!
</Example>
@ -1393,15 +1399,23 @@ export default class Statistica extends Component {
</Panel>
</Split>
<Split>
<Panel title={"Una gamma particolare"}>
<Panel title={"Gamma e normale"}>
<p>
La distribuzione normale ha una particolare relazione con la distribuzione Gamma:
</p>
<p>
<Latex>{r`Z^2 \sim \chi^2 (v = 1)`}</Latex>
</p>
</Panel>
<Panel title={"La funzione Chi"}>
<blockquote>
chi-quadro a un grado di libertà
"chi-quadro a un grado di libertà"
</blockquote>
<p>
Esiste una distribuzione Gamma particolare:
</p>
<p>
<Latex>{r`\Gamma (\frac{1}{2}, \frac{1}{2}) = \chi^2 (v = 1)`}</Latex>
<Latex>{r`\Gamma \left( \frac{1}{2}, \frac{1}{2} \right) = \chi^2 (v = 1)`}</Latex>
</p>
<p>
Più chi-quadro possono essere sommate per aumentare i loro gradi di libertà:
@ -1410,12 +1424,12 @@ export default class Statistica extends Component {
<Latex>{r`\chi^2 (n) + \chi^2 (m) = \chi^2 (n + m)`}</Latex>
</p>
</Panel>
<Panel title={"Gamma e normale"}>
<Panel title={"T di Student"}>
<p>
La distribuzione normale ha una particolare relazione con la distribuzione Gamma:
Un'altra funzione particolare è la funzione T di Student:
</p>
<p>
<Latex>{r`Z^2 \sim \chi^2 (v = 1)`}</Latex>
<Latex>{r`T(v) = \frac{Nor(0, 1)}{\sqrt{\frac{\chi^2(v)}{v}}}`}</Latex>
</p>
</Panel>
</Split>
@ -1807,6 +1821,9 @@ export default class Statistica extends Component {
<p>
<Latex>{r`\forall \epsilon > 0, P \left( \lim_{n \to +\infty} | \overline{X}_n - E(X) | < \epsilon \right) = 1`}</Latex>
</p>
<Example>
Dimostra che l'interpretazione frequentista della probabilità è valida!
</Example>
</Panel>
</Split>
<Split title={"Al limite"}>
@ -2058,7 +2075,15 @@ export default class Statistica extends Component {
</Panel>
<Panel title={"Varianza incognita"}>
<p>
<Todo>TODO: Cos'è la distribuzione di Student?</Todo>
Se non conosciamo la varianza di una normale, allora possiamo ricavare velocemente gli intervalli di confidenza all'<Latex>\alpha</Latex>% con queste formule:
</p>
<ul>
<li>Intervalli bilateri: <Latex>{r`\mu \in \left[ \overline{x}_n - t_{1 - \frac{\alpha}{2}; n-1} \cdot \sqrt{\frac{s_n^2}{n}}, \overline{x}_n + t_{1 - \frac{\alpha}{2}; n-1} \cdot \sqrt{\frac{s_n^2}{n}} \right]`}</Latex></li>
<li>Intervallo unilatero da sinistra: <Latex>{r`\mu \in \left( -\infty, \overline{x}_n + t_{1 - \frac{\alpha}{2}; n-1} \cdot \sqrt{\frac{s_n^2}{n}} \right]`}</Latex></li>
<li>Intervallo unilatero da destra: <Latex>{r`\mu \in \left[ \overline{x}_n - t_{1 - \frac{\alpha}{2}; n-1} \cdot \sqrt{\frac{s_n^2}{n}}, +\infty \right)`}</Latex></li>
</ul>
<p>
<Latex>{r`t_{\alpha, v}`}</Latex> è un quantile della distribuzione di Student di parametro <Latex>v</Latex>.
</p>
</Panel>
</Split>