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1. Definitio et divisio temere variabilium

Definitio: Temere variabilis variabilis est quae varias valores in experimento temere accipit.

1. Categorica temere variabilis (Nominal)

   • Exempli gratia: Gender (masculus, femina), occupatio (servus civilis, operarius corporatus, discipulus, recessus, otiosus), effectus test (negativus, affirmativus)

2. Categorica temere variabilis iussit (Iussit)

   • Exempla: habitus (vehementer adsentior, consentio, neuter, abeo, valde dissentio), frequentatio usus (semel in hebdomade, semel singulis quindecim, semel singulis sex mensibus, fere numquam, numquam);

3. Numeralis temere variabilis

   • Exemplum: Aetas (13, 14, 15, 16...), reditus (quovis valore implere potes)

Probabilitas 2. discrete distribution

2.1 Binomial Distributio

• definition: descriptus in $n$ Prosperum in iuris iudiciis$k$ Probabilitas victoria in se iudicium est$p$。
• formula： $X\sim \text{Bin}(n,p)$
• Probabilitas Missae Function (PMF)： $P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$
• Munus cumulativum distributionis (CDF)： $F(X=k)={\sum }_{ego=0}^k\left(\genfrac{}{}{0ex}{}{n}{ego}\right)p^{ego}(1-p{)}^{n-ego}$

2.2 Bernoullius Distributio

• definition: Probabilitas victoria describitur (vel defectum) in experimentum $p$。
• formula：$X\sim \text{Bern}(p)$
• Probabilitas Missae Function (PMF)： $P(X=x*******************************************)=\{\begin{array}{ll}p& \text{si}x*******************************************=1\\ 1-p& \text{si}x*******************************************=0\end{array}$
• Munus cumulativum distributionis (CDF)： $F(X=x*******************************************)=\{\begin{array}{ll}0& \text{si}x*******************************************<0\\ 1-p& \text{si}0\le x*******************************************<1\\ 1& \text{si}x*******************************************\ge 1\end{array}$

2.3 Distributio Geometrica

• definition: Probabilitas numeri defectorum ante primam successum describit. Probabilitas successus in unoquoque iudicio est $p$。
• formula： $X\sim \text{Geom}(p)$
• Probabilitas Missae Function (PMF)： $P(X=k)=(1-p{)}^{k}p$
• Munus cumulativum distributionis (CDF)： $F(X=k)=1-(1-p{)}^{k+1}$

2.4 Negative Binomial Distributio

• definition: Defectus numerum describit antequam successu rth. Probabilitas successus pro unoquoque iudicio est $p$。
• formula： $X\sim \text{NegBin}(r*****,p)$
• Probabilitas Missae Function (PMF)： $P(X=k)=\left(\genfrac{}{}{0ex}{}{k+r*****-1}{k}\right)p^{r*****}(1-p{)}^k$
• Munus cumulativum distributionis (CDF)： $F(X=k)={\sum }_{ego=0}^k\left(\genfrac{}{}{0ex}{}{ego+r*****-1}{ego}\right)p^{r*****}(1-p{)}^{ego}$

2.5 Hypergeometrica Distributio

• definitionDescribitur sampling ab hominibus finitis sine replacement $n$ temporibus secundis$k$ temporibus probabilius.
• formula： $X\sim \text{Hypergeom}(N,K,n)$
• Probabilitas Missae Function (PMF)： $P(X=k)=\frac{\left(\genfrac{}{}{0ex}{}{K}{k}\right)\left(\genfrac{}{}{0ex}{}{N-K}{n-k}\right)}{\left(\genfrac{}{}{0ex}{}{N}{n}\right)}$
• Munus cumulativum distributionis (CDF)： $F(X=k)={\sum }_{ego=0}^k\frac{\left(\genfrac{}{}{0ex}{}{K}{ego}\right)\left(\genfrac{}{}{0ex}{}{N-K}{n-ego}\right)}{\left(\genfrac{}{}{0ex}{}{N}{n}\right)}$

2.6 Poisson Distribution

• definition: Descriptio fit in unitatis tempore $k$ Probabilitas rei, mediocris eventus, eventus est λ.
• formula： $X\sim \text{Poisson}(\lambda )$
• Probabilitas Missae Function (PMF)： $P(X=k)=\frac{{\lambda }^k{e}^{-\lambda }}{k!}$
• Munus cumulativum distributionis (CDF)： $F(X=k)=e^{-\lambda }{\sum }_{ego=0}^k\frac{{\lambda }^{ego}}{ego!}$

2.7 Relatio inter eos

• Bernoullius distributionest specialisbinomialis distribution, cum $n=1$ horam.
• geometrica distributioneest distributio numerorum iudiciorum ante primam successum requisiti, qui videri potestbinomialis distributionprorogatio.
• negans binomium distributionvideri potest quodgeometrica distributioneGeneralizationis, numerum defectuum ante r successuum requisitum describere.
• hypergeometrica distributionSimilia tobinomialis distributionsed aptas ad finitimas nationes ac sampling non reponenda.
• Distributio Poissonsicbinomialis distributionFinis casus $n$ valde magnum et$p$ infantulus, and$\lambda =np$ assidue serva.

3. continua probabilitas distribution

3.1 Exponentialis Distributio

• definitionDistributio exponentialis est continua probabilitas distributio saepe ad describendum tempora intervalla inter eventus independentes.

• formula： $X\sim \text{Exponentialis}(\lambda )$,in $\lambda >0$ est rate parametri

• Probabilitas Density Function (PDF)：$f******(x*******************************************;\lambda )=\lambda e^{-\lambda x*******************************************}\text{for*}x*******************************************\ge 0$

• Munus cumulativum distributionis (CDF)：$F(x*******************************************;\lambda )=1-e^{-\lambda x*******************************************}\text{for*}x*******************************************\ge 0$

3.2 Gamma Distribution

• definition: Describitur tempus exspectationis cumulum et est generalisatio distributionis exponentiae et χ² distributio.

• formula： $X\sim \text{Gamma}(k,\theta )$,in $k>0$ figura parametri;$\theta >0$ scala parametri

• Probabilitas Density Function (PDF)：$f******(x*******************************************;k,\theta )=\frac{{x*******************************************}^{k-1}{e}^{-x*******************************************/\theta }}{{\theta }^k\Gamma (k)}\text{for*}x*******************************************\ge 0$,in $\Gamma (k)$ gamma munus est

• Munus cumulativum distributionis (CDF)：$F(x*******************************************;k,\theta )=\frac{\gamma (k,x*******************************************/\theta )}{\Gamma (k)}$,in $\gamma (k,x*******************************************/\theta )$ gamma munus incompletum est.

3.3 Normal Distribution

• definition: Distributio summae describitur permagnum numerum independens passim variabilium variarum et late in scientiis naturalibus et in scientiis socialibus.

• formula：$X\sim \mathcal{N}(\mu ,{\sigma }^2)$, in,$\mu$ vilis pretii;${\sigma }^2$ sit dissidio

• Probabilitas Density Function (PDF)：$f******(x*******************************************;\mu ,{\sigma }^2)=\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{(x*******************************************-\mu {)}^2}{2{\sigma }^2}}$

• Munus cumulativum distributionis (CDF)：$F(x*******************************************;\mu ,{\sigma }^2)=\frac{1}{2}\left[1+erf*\left(\frac{x*******************************************-\mu }{\sqrt{2{\sigma }^2}}\right)\right]$, ubi erroris munus erbe.

3.4 t-Distribution

• definition: usus pro hypothesi probatio et fiducia inter- estimatio in parvis casibus specimen.

• formula： $X\sim t(\nu )$, in,$\nu$ gradus libertatis

• Probabilitas Density Function (PDF)：$f******(t;\nu )=\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\sqrt{\nu \pi }\Gamma \left(\frac{\nu }{2}\right)}{\left(1+\frac{t^2}{\nu }\right)}^{-\frac{\nu +1}{2}}$

• Munus cumulativum distributionis (CDF)：$F(t;\nu )=\frac{1}{2}+t\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\sqrt{\pi \nu }\Gamma \left(\frac{\nu }{2}\right)}{\int }_0^t{\left(1+\frac{{u**}^2}{\nu }\right)}^{-\frac{\nu +1}{2}}d************************u**$

3.5 Chi-Square Distributio

• definition: Communiter pro hypothesi probatio et analysis variantis.

• formula： $X\sim {\chi }^2(k)$,in $k$ gradus libertatis

• Probabilitas Density Function (PDF)：$f******(x*******************************************;k)=\frac{1}{2^{k/2}\Gamma (k/2)}{x*******************************************}^{k/2-1}{e}^{-x*******************************************/2}\text{for*}x*******************************************\ge 0$

• Munus cumulativum distributionis (CDF)：$F(x*******************************************;k)=\frac{\gamma \left(\frac{k}{2},\frac{x*******************************************}{2}\right)}{\Gamma \left(\frac{k}{2}\right)}$

3.6 F-Distribution

• definition: duo exempla comparare solebant.

• formula： $X\sim F({d************************}_1,{d************************}_2)$,in ${d************************}_1$ et${d************************}_2$ gradus libertatis

• Probabilitas Density Function (PDF)：$f******(x*******************************************;{d************************}_1,{d************************}_2)=\frac{\sqrt{\frac{({d************************}_{1}x*******************************************{)}^{{d************************}_1}{d************************}_2^{{d************************}_2}}{({d************************}_{1}x*******************************************+{d************************}_2{)}^{{d************************}_1+{d************************}_2}}}}{x*******************************************B\left(\frac{{d************************}_1}{2},\frac{{d************************}_2}{2}\right)}\text{for*}x*******************************************\ge 0$,in $B$ est beta munus

• Munus cumulativum distributionis (CDF)：$F(x*******************************************;{d************************}_1,{d************************}_2)={ego}_{\frac{{d************************}_{1}x*******************************************}{{d************************}_{1}x*******************************************+{d************************}_2}}\left(\frac{{d************************}_1}{2},\frac{{d************************}_2}{2}\right)$,in $ego$ est incompleta beta munus

3.7 Relatio inter eos

• Chi-quadratus distributio estQuadratorum summa distributio normalis . Exempli gratia$k$ Summa quadratorum ex normalibus differentiis independens gradibus libertatis obedit$k$ quadra- chi distributio.
• Distributio est inConstructum ex normali distributione vexillum et distributionem chi-quadratum of. Speciatim distributio t-distributionis obtineri potest dividendo vexillum normale variabile secundum radicem quadratam sui iuris divulgationis chi-quadratae variabilis.
• F est distributioExtensio rationis duorum independens chi-quadratorum variabilium distributarum . F distributio adhibetur ad binas differentias comparandas et construitur sumendo rationem duarum variabilium chi-quadratorum distributarum.