#exemple 7.2d

x=c(4,0,6,5,2,1,2,0,4,3)
lambda=mean(x)
ppois(2,lambda)

#exemple 7.2f

y=c(2.2,3.4,1.6,0.8,2.7,3.3,1.6,2.8,2.5,1.9)
x=log(y)
mu=mean(x)
sigma=sd(x)
pnorm(log(3),mu ,sigma)-pnorm(log(2),mu ,sigma)


#exemples 7.3f

x=c(54,63,58,72,49,92,70,73,69,104,48,66,80,64,77)
t.test(x,alternative="two.sided",conf.level=.95)
t.test(x,alternative="less",conf.level=.95)


#exemple 7.3g, vrai theta est pi/4=0.7854

n=100
u=runif(n)
x=sqrt(1-u^2)
t.test(x,alternative="two.sided",conf.level=.95)

n=100000 #pour plus de precision
u=runif(n)
x=sqrt(1-u^2)
t.test(x,alternative="two.sided",conf.level=.99)

#integrable double sur le disque unite de sqrt[1-(u1^2+u2^2)] #vraie valeur de 2 pi/3=2.0944
n=100000
u1=runif(n)
u2=runif(n)
y=u1^2+u2^2
y=y[y<1]
x=pi*sqrt(1-y)
t.test(x,alternative="two.sided",conf.level=.99)

#exemple 7.3h

x=c(.123,.124,.126,.12,.13,.133,.125,.128,.124,.126)
n=length(x)
ic=(n-1)*var(x)/qchisq(c(.95,.05),n-1)
ic #intervalle pour sigma^2
sqrt(ic) #intervalle pour sigma

#exemple 7.4b

x=c(140,136,138,150,152,144,132,142,150,154,136,142)
y=c(144,132,136,140,128,150,130,134,130,146,128,131,137,135)
t.test(x,y,alternative="two.sided",var.equal=TRUE,conf.level=.9)
t.test(x,y,alternative="greater",var.equal=TRUE,conf.level=.95)

#exemple 7.5a

#exact
binom.test(80,100,alternative="two.sided",conf.level=.95)

#approximation normale
p=.8
n=100
alpha=.05
z=qnorm(1-alpha/2)
p+c(-1,1)*z*sqrt(p*(1-p)/n)

#exemple 7.6a

somme=1740
n=10
ic=2*somme/qchisq(c(.975,.025),2*n)
ic


#exercice 29

vecN=NULL
for(i in 1:10000){
N=0
sum=0
while(sum<1){
sum=sum+runif(1)
N=N+1}
vecN=c(vecN,N)
}
t.test(vecN,alternative="two.sided",conf.level=.99)