# Section 5; Question 1:
#
p = 0.3
r = 4
xs = r:7
ps = choose( xs - 1, r - 1 ) * p^r * (1-p)^(xs-r)
print(sum(ps))


# Section 5; Question 2:
#
p = 0.3
r = 3
xs = 7
ps = choose( xs - 1, r- 1 ) * p^r * (1-p)^(xs-r)
print( sum( ps ))


# Section 5; Question 3:
#
source('chap_4_sect_5_question_3_data.R')
T = table(data)
empirical = T/length(data)
ks = min(data):max(data)
r = 2
p = 1/2
model = choose( ks-1, r-1) * p^r * (1-p)^(ks-r)
print( data.frame( k=ks, empirical=as.double( empirical ), model=as.double( round( model, 2 ) ) ) )


# Section 5; Question 4:
#
r = 3
p = 0.15
ks = r:4 # compute P(X<5) first
p_x_lt_5 = sum( choose( ks-1, r-1 ) * p^r * (1-p)^(ks-r) )
p_x_ge_5 = 1 - p_x_lt_5
print( c( p_x_ge_5, r/p ) )


# Section 5; Question 8:
#
r = 9
p = 4/5
ks = r:12 # compute P(X <= 12) first
p_x_lt_12 = sum( choose( ks-1, r-1 ) * p^r * (1-p)^(ks-r) )
ks = r:9  # compute P(X <= 9) next 
p_x_lt_9 = sum( choose( ks-1, r-1 ) * p^r * (1-p)^(ks-r) )
print( p_x_lt_12 - p_x_lt_9 )