#(1)
library(quantreg)
Y=c(2,0,-5,5,1)

#
ind=seq(0,1, by=0.01) #graf vykreslime na mrizce od 0 do 1
quantile(Y,ind,type=1)

par(mfrow=c(3,3))
plot(quantile(Y,ind,type=1)~ind,type='l')
plot(quantile(Y,ind,type=2)~ind,type='l')
plot(quantile(Y,ind,type=3)~ind,type='l')
plot(quantile(Y,ind,type=4)~ind,type='l')
plot(quantile(Y,ind,type=5)~ind,type='l')
plot(quantile(Y,ind,type=6)~ind,type='l')
plot(quantile(Y,ind,type=7)~ind,type='l')
plot(quantile(Y,ind,type=8)~ind,type='l')
plot(quantile(Y,ind,type=9)~ind,type='l')




rq(Y~1,tau=0)
rq(Y~1,tau=0.01)
rq(Y~1,tau=0.25)
rq(Y~1,tau=0.6)
rq(Y~1,tau=0.8)
rq(Y~1,tau=1)

rq(Y~1,tau=ind)
rq(Y~1,tau=-1)

plot(rq(Y~1,tau=-1))


#(a)
R=rank(Y)
R

#(b)
P=rq(Y~1, tau=-1)$sol   
D=rq(Y~1, tau=-1)$dsol  #y-ove souradnice zlomu
ind=P[1,]               #x-ove souradnice zlomu 

par(mfrow=c(2,3))
for (i in 1:5){
plot(D[i,]~ind,type="l", main=c("R=",R[i]),xlab=expression(alpha),ylab="a_i")
}


#(2)
library(HSAUR)
data=heptathlon
attach(data)


M=rq(score~hurdles+highjump+longjump+javelin,data=heptathlon, tau=-1)

rq(score~hurdles+highjump+longjump+javelin,data=heptathlon, tau=0.49)
rq(score~hurdles+highjump+longjump+javelin,data=heptathlon, tau=0.51)

#(a)
plot(M)  #regresni kvantilovy proces 

P=M$sol       #y-ove souradnice bodu zlomu
ind=P[1,]     #x-ove souradnice bodu zlomu

par(mfrow=c(3,2)) #vlastni konstrukce tehoz
for (i in 4:8){
plot(P[i,]~ind)
}

#(b)
#nejprve definujeme model s 9 pozorovanimi
M2=rq(score~hurdles+highjump+longjump+javelin,data=heptathlon[1:9,], tau=-1)

P2=M2$sol
ind2=P2[1,]     #x-ove souradnice bodu zlomu
D2=M2$dsol      #y-ove souradnice bodu zlomu

par(mfrow=c(3,3))
for (i in 1:9){
plot(D2[i,]~ind2,type="l",xlab=expression(alpha),ylab="a_i")
}


#(3)
m=5
n=7
N=m+n

#(a)
X=combn(N,n)
K=length(X[1,])

W=apply(X,2,sum)
table(W)/K        #pravdepodobnostni funkce
plot(table(W)/K,xlab="x",ylab="p(x)")  #graf pravdepodobnostni funkce

#(b)
mu=n*(N+1)/2             #EW
sigma=sqrt(m*n*(N+1)/12) #sd(W)
curve(dnorm(x,mu,sigma),add=T,col=2) #graf as. normalniho rozdeleni

#(c)
W2=W-n*(n+1)/2
table(W2)/K
dwilcox(0:35,m,n)