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Distribución muestral de \(S^{2}\).

Sea \(X_{1}, X_{2}, X_{3},\ldots,X_{n}\) una muestra aleatoria de una distribución normal con media \(\mu\) y varianza \(\sigma^{2}\). La variable aleatoria:

\[\begin{equation*} \dfrac{(n-1)S^{2}}{\sigma^{2}}=\dfrac{\sum(X_{i}-\bar{X})^{2}}{\sigma^{2}} \end{equation*}\]

tiene una una distribución chi-cuadrado co \(n-1\) grados de libertad.

Ilustración.

Sea la población:

w<-c(293.9,103.0,200.5,248.1,85.1,242.5,322.7,22.0,242.4,216.8,130.8,163.9,132.9,76.1,240.6,162.8,153.9,154.9,192.8,213.7,295.6,60.0,328.5,195.9,145.5,199.7,157.5,199.2,157.1,320.4,91.0,332.2,249.8,148.2,183.3,242.6,164.2,169.7,251.2,308.6,217.4,59.1,103.2,289.3,153.6,305.6,283.8,169.7,159.3,128.0,207.7,233.9,301.2,125.8,164.2,371.6,275.1,85.7,241.9,429.3,178.3,286.0,274.3,288.1,287.4,181.4,185.7,372.9,144.8,221.7,168.8,171.9,210.1,169.4,199.7,192.7,243.6,212.5,164.4,142.2,146.9,146.6,151.3,267.5,152.1,251.5,153.0,320.1,119.3,193.3,74.8,252.1,249.8,169.8,162.5,240.7,190.1,320.1,210.8,198.9,125.2,260.4,258.9,247.1,269.7,160.8,205.3,208.7,301.5,189.1,327.4,204.2,246.6,209.3,224.9,226.1,173.7,281.3,154.9,210.6,81.0,192.8,249.8,196.6,143.3,48.6,306.0,206.9,217.3,263.2,179.2,264.3,350.2,255.8,197.1,156.8,125.5,60.3,105.3,137.8,237.1,170.6,117.0,154.6,209.2,213.9,273.9,316.8,142.8,187.8,280.1,158.8,310.1,249.1,63.1,320.0,245.9,194.4,258.6,278.5,145.2,235.6,224.5,61.3,310.3,152.4,216.4,133.0,282.0,158.9,258.5,113.5,215.0,196.5,236.0,199.5,212.5,312.8,93.2,327.8,214.1,292.4,118.2,234.1,154.0,258.3,143.9,168.9,283.5,228.3,139.1,269.7,128.1,310.9,250.3,247.8,268.0,270.3,98.0,206.2)

Histograma de la población

hist(w,prob=TRUE)

n<-length(w);n
## [1] 200

Varianza de la población.

vapo<-var(w)*(n-1)/n;vapo
## [1] 5238.815

Seleccionar 10000 muestras aleatorias con reemplazamiento de tamaño 5.

mue<-lapply(1:10000,function(x)
sample(w,5,replace=TRUE))
vari<-sapply(mue,var)

Estadístico chi-cuadrado.

ch<-4*vari/vapo
hist(ch,prob=TRUE,col="yellow")
curve(dchisq(x,4),xlim=c(0,50),add=TRUE,lwd=2)

Seleccionar 10000 muestras aleatorias con reemplazamiento de tamaño 10.

mue1<-lapply(1:10000,function(x)
sample(w,10,replace=TRUE))
vari1<-sapply(mue1,var)

Estadístico chi-cuadrado.

ch1<-9*vari1/vapo
hist(ch1,prob=TRUE,col="yellow")
curve(dchisq(x,9),xlim=c(0,50),add=TRUE,lwd=2)

Seleccionar 10000 muestras aleatorias con reemplazamiento de tamaño 20.

mue2<-lapply(1:10000,function(x)
sample(w,20,replace=TRUE))
vari2<-sapply(mue2,var)

Estadístico chi-cuadrado.

ch2<-19*vari2/vapo
hist(ch2,prob=TRUE,col="yellow",ylim=c(0,0.08))
curve(dchisq(x,19),add=TRUE,lwd=2,xlim=c(0,50))

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O.M.F.

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