3.6

#1a
dgeom(5, prob = 0.04)

[1] 0.03261491

#1b
1 - pgeom(7, prob = 0.04)

[1] 0.7213896

#1e
n <- 10000
p <- 0.04

x <- rgeom(n, prob = p)
y <- x + 1

mean(y == 6)

[1] 0.0341

mean(x >= 8)

[1] 0.7076

mean(x)

[1] 23.9627

mean(0.45 * y)

[1] 11.23322

#2a
dnbinom(4, size = 3, prob = 0.2)

[1] 0.049152

#2b
1 - pnbinom(6, size = 3, prob = 0.2)

[1] 0.7381975

#2c
qnbinom(0.4, size = 3, prob = 0.2) + 3

[1] 12

#2e
n <- 10000
x <- rnbinom(n, size = 3, prob = 0.2)
y <- x + 3

mean(y == 7)

[1] 0.0492

mean(y >= 10)

[1] 0.7418

quantile(y, 0.4)

40% 
 12 

mean(y)

[1] 14.9928