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Problem 3.2

p=c(1/4,1/2,1/8,1/8)
x=c(0:3)
s=sample(x,10000,replace=TRUE,prob=p) 
one=s[s==1] 
length(one)/length(s) 
## [1] 0.5065
t=table(s)
prop.table(t)
## s
##      0      1      2      3 
## 0.2510 0.5065 0.1218 0.1207

Problem 3.34

Set1 <- rpois(300,2)

Set2 <- dpois(Set1,2)

Set3 <- choose(300,Set2)*(2/300)^Set2*(1-2/300)^(300-Set2)

Set4 <- Set3 - Set2

max(Set4)
## [1] 0.1287026

Problem 4.13

X <- rnorm(10000, 0, 2)
Y <- rnorm(10000, 0, 1)
Z <- X+ 3*Y
var(Z)
## [1] 13.19252
C <- 1^2*4+3^2*1
## var(Z) > C
event <- replicate(10000, {
  dice <- sample(x = 1:6, size = 7, replace = TRUE) # roll seven dice
  sum(dice) > 30 # test if the event occurred
})
mean(event)
## [1] 0.096