Esempio: distribuzione di Cauchy

Funzioni > Statistica > Distribuzioni di probabilità > Esempio: distribuzione di Cauchy

1. Mostrare la definizione della distribuzione di Cauchy:

dove:

◦ l è il parametro di posizione (media)

◦ s è il parametro di scala (radice quadrata della varianza), s > 0

2. Definire tre insiemi di parametri di posizione e di scala e quindi calcolare l'ampiezza, o altezza, della curva con i parametri di posizione e di scala l0 e s0:

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3. Utilizzare la funzione dcauchy per ottenere la densità di probabilità per il valore x utilizzando diversi valori per i parametri di posizione e un parametro di scala fisso:

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Diversi valori per i parametri di posizione spostano la curva lungo l'asse x.

4. Tracciare il grafico della funzione dcauchy utilizzando un parametro di posizione fisso e diversi valori per i parametri di scala:

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Diversi valori per i parametri di scala modificano l'altezza della curva.

5. Tracciare il grafico della funzione dcauchy utilizzando diversi valori per i parametri di posizione e di scala:

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 <yAxis rank="1" legend-position="PlotBoundaryLeft" start="0" end="0.33">
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 <axisGrid>
 <gridFrequency>12</gridFrequency>
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 <gridLines />
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 <markers>
 <marker resultRef="324" display="true" color="#FF2E3192" line-style="Dash" value="0.1061032953945969">
 <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">
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 </math>
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Diversi valori per i parametri di scala e di posizione spostano le curve e ne modificano l'altezza.

6. Tracciare il grafico della funzione pcauchy utilizzando diversi valori per i parametri di posizione e un parametro di scala fisso:

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 <id xml:space="preserve">
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 <math resultRef="355">
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◦ Diversi valori per i parametri di posizione spostano la curva lungo l'asse x.

◦ Tutti i valori y di pcauchy sono compresi tra 0 e 1.

7. Tracciare il grafico della funzione pcauchy utilizzando un parametro di posizione fisso e diversi valori per i parametri di scala:

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 <id xml:space="preserve">
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◦ Diversi valori per i parametri di scala appiattiscono la curva in orizzontale, ma tutte le curve si incrociano in corrispondenza di y=0.5.

◦ Tutti i valori y di pcauchy sono compresi tra 0 e 1.

8. Tracciare il grafico della funzione qcauchy utilizzando diversi valori per i parametri di posizione e un parametro di scala fisso:

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 <apply xmlns="http://schemas.mathsoft.com/math50">
 <id labels="FUNCTION" xml:space="preserve">qcauchy</id>
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 <id xml:space="preserve">
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 <id xml:space="preserve">
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 <apply xmlns="http://schemas.mathsoft.com/math50">
 <id labels="FUNCTION" xml:space="preserve">qcauchy</id>
 <sequence>
 <id xml:space="preserve">x</id>
 <id xml:space="preserve">
 l<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">1</Subscript>
 </id>
 <id xml:space="preserve">
 s<Subscript xmlns="clr-namespace:Ptc.Wpf;assembly=Ptc.Core">0</Subscript>
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 <id xml:space="preserve">
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 <id xml:space="preserve">
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◦ Diversi valori per i parametri di posizione spostano la curva lungo l'asse y.

◦ Tutti i valori x di qcauchy sono compresi tra 0 e 1.

9. Calcolare la funzione rcauchy utilizzando parametri di posizione e di scala fissi:

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La funzione rcauchy restituisce un vettore di m numeri casuali con distribuzione di Cauchy.

10. Tracciare il grafico dei numeri casuali restituiti dalla funzione rcauchy.

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 </math>
 <math resultRef="454">
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </math>
 </plotEquation>
 </plotEquations>
 <xyDomain scale-type="linear" auto-scale="true">
 <startValue>
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </startValue>
 <endValue>
 <placeholder xmlns="http://schemas.mathsoft.com/math50" />
 </endValue>
 </xyDomain>
 </yAxis>
 </axes>
 </xyPlot>
 </plot>
</region>

Se si ricalcola il foglio di lavoro, la funzione rcauchy restituisce un nuovo insieme di numeri casuali e il grafico viene aggiornato di conseguenza.

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