Distribución Beta

Se trabajan unos datos de proporción de germinación con una distribución beta. Total de datos: 80, parámetros a y b, 5 y 0,2 respectivamente.

set.seed(2022)
pgerm=rbeta(80,5,0.2)
mean(pgerm)

## [1] 0.9657285

hist(pgerm)
abline(v=mean(pgerm),lty=2,col="red",lwd=2)

var(pgerm)

## [1] 0.004259226

Valor esperado

\[E[X]=\frac{\alpha}{\alpha+\beta}=0.9657285\]

\[\alpha = 0.9657285(\alpha+\beta)\]

\[\alpha = 0.9657285\alpha+0.9657285\beta \]

\[\alpha- 0.9657285\alpha= 0.9657285\beta \]

\[0.0342715\alpha= 0.9657285\beta \]

\[\alpha= 28.17876\beta \]

Varianza

\[Var[X]=\frac{\alpha\beta}{(\alpha+\beta+1)( \alpha+\beta)^2}=0.004259226\]

\[\frac{28.17876\beta ^2}{(28.17876\beta +\beta+1)( 28.17876\beta +\beta)^2}=0.004259226\]

\[\frac{28.17876\beta ^2}{(29.17876\beta+1)( 29.17876\beta)^2}=0.004259226\]

\[\require{cancel} \frac{28.17876\cancel{\beta ^2}}{(29.17876\beta+1)851.4\cancel{\beta^2}}=0.004259226\]

\[\require{cancel} \frac{28.17876} {(29.17876\beta+1)851.4}=0.004259226\]

\[\require{cancel} \frac{28.17876} {24842.79626\beta+851.4}=0.004259226\]

\[28.17876=0.004259226*(24842.79626\beta+851.4)\]

\[28.17876=105.8111\beta+3.6263\]

\[24.552=105.8111\beta\]

\[\beta=\frac{24.552}{105.8111}\]

\[\beta=0.232\]