Chapter 7, page 290, Question 7.

a.) A die is rolled three times with outcomes \(X_1\), \(X_2\), and \(X_3\). Let \(Y_3\) be the maximum of the values obtained. Show that \(P(Y_3 \leq j) = P(X_1 \leq j)^3\).

Because the \(X_i\)s are from the same die, they all have the same probability. Therefore \(P(X_1)\leq j)=P(X_2\leq j)=P(X_3\leq j)\). \(Y_3\) is the minimum of the X’s, so it is checking whether all three are less than \(j\), that is \(P(Y_3\leq j)=P(X_1\leq j)\times P(X_2\leq j)\times P(X_3 \leq j)= P(X_1\leq j)^3\).

Use this to find the distribution of \(Y_3\). Does \(Y_3\) have a bell-shaped distribution?

\(Y_3\) does not have a bell-shaped distribution. Probabilities are being multiplied, not added here, so we do not get the same shape for the distribution as we would if we added the rolls of a die. To verify this in R:

y<-min(sample(1:6,3,replace = T))
for(i in 1:100000){
  y<-cbind(y,min(sample(1:6,3,replace = T)))  
}
hist(y)

y2<-t(y)
summary(y2)
##        V1       
##  Min.   :1.000  
##  1st Qu.:1.000  
##  Median :2.000  
##  Mean   :2.042  
##  3rd Qu.:3.000  
##  Max.   :6.000

b.) Now let \(Y_n\) be the maximum value when \(n\) dice are rolled. Find the distribution of \(Y_n\). Is this distribution bell-shaped for large values of \(n\)?

For any large value of n, the distribution takes on essentially a single value. In R:

w<-max(sample(1:6,12,replace = T))
for(i in 1:100000){
  w<-cbind(w,max(sample(1:6,12,replace = T)))  
}
w2<-t(w)
summary(w2)
##        V1      
##  Min.   :3.00  
##  1st Qu.:6.00  
##  Median :6.00  
##  Mean   :5.88  
##  3rd Qu.:6.00  
##  Max.   :6.00