# Exercise 3.5 Binomial Example n=100, p=0.8
x=seq(85,100,by=1)
prob=choose(100,x)*(.8)^x*(.2)^{100-x}
sum(prob)
# using dbinom function
sum(dbinom(x,100,0.8))
# with Normal approximation
n=100;p=0.8
mu=n*p
sd=sqrt(n*p*(1-p))
z=(85-mu)/sd
1-pnorm(z)
# with Poisson approximation (works bad!)
lambda=n*p
sum(dpois(x,lambda))
# Exercise 3.7
# set lambda values
lam=seq(0,10,by=0.1)
prob=exp(-lam) + lam*exp(-lam)
# smallest value of lam
min(lam[prob<=.01])