You can sum vectors in R using the command sum(). For example, to sum the probabilities from Y ∼ Binomial(10, 0.5)

from Y = 2 to Y = 7, the command is sum( dbinom(2:7, 10, 0.5) )

For each of the probabilities below, give the appropriate R command two ways: firstly using pbinom(), and secondly using sum(dbinom()). Run the commands in R and show they give the same answers.

a. P(Y ≤ 12) when Y ∼ Binomial(15, 0.7).

b. P(Y < 30) when Y ∼ Binomial(50, 0.6).

c. P(Y > 72) when Y ∼ Binomial(80, 0.9).

d. P(60 ≤ Y ≤ 70) when Y ∼ Binomial(100, 0.6).

e. P(15 < Y < 20) when Y ∼ Binomial(30, 0.5)