# PROVA FINALE 3 LDS -- febbraio 2025 # ES. 1 rm(list = ls()) al=5 be=3 n=20 sn=12 al.p=al+sn be.p=be+n-sn M=10000 th.mc=rbeta(M,al.p,be.p) lam.mc=log(th.mc/(1-th.mc)) mean(lam.mc) # 0.44 var(lam.mc) # 0.15 # Es. 2 rm(list = ls()) al=4 be=2 n=8 sn=20 al.p=al+sn be.p=be+n gamma=0.1 M=10000 th.mc=rgamma(M,al.p,rate=be.p) psi.mc=1/sqrt(th.mc) L=quantile(psi.mc,gamma/2) U=quantile(psi.mc,1-gamma/2) c(L,U) # 0.56 - 0.78 # Es. 3 rm(list = ls()) al=1 be=5 delta=1.7 lambda=1.3 M=10000 w.mc=rgamma(M,al,rate=be) mean(w.mc) loss.mc=abs(w.mc-delta) # mean(loss.mc) mean(loss.mc>lambda) # 0.88 # Es. 4 rm(list = ls()) al=3 be=4 n=12 M=10000 th.mc=rgamma(M,al,rate=be) cn=(2*n-1)/(n^2) R.mc=cn*th.mc^2 mean(R.mc) # 0.12 # Es. 5 rm(list = ls()) al=5 be=4 M=100000 x.mc=rgamma(M,al,rate=be) E.x=mean(x.mc) E.x mu3.mc=mean((x.mc-E.x)^3) mu3.mc # 0.15 # Es. 6 rm(list = ls()) a=0 b=1.3 n=10 M=10000 theta=4 x.matrice=runif(M*n,min=0,max=theta) x.matrice=matrix(x.matrice,M,n) X.max=apply(x.matrice,1,max) L=X.max+a U=X.max+b mean(X.max) mean(L<=theta&U>=theta) # 0.98 # Es. 7 rm(list = ls()) M=50000 th.0=3 th.1=2 n=10 k=2.2 # Per il calcolo di alpha (simulare sotto H0) x.matr.H0=rnorm(n*M,mean=0,sd=sqrt(th.0)) x.matr.H0=matrix(x.matr.H0,M,n) S.H0=apply(x.matr.H0,1,var) b=(n-1)/(n+1) d=b*S.H0 mean(d