#  Example code for finding Poisson probabilities in R

#  To get the cumulative probability, the syntax is: rpois(x, lambda) 
#  where lambda and x are values that you fill in 

# Example 1 from class notes on the Poisson:
# X is Poisson with lambda=1.

# P(X <= 1):
ppois(1, lambda=1)

# P(X >= 3) = 1 - P(X <= 2)
1 - ppois(2, lambda=1)

# P(X=2) = P(X <=2) - P(X<=1):
ppois(2, lambda=1) - ppois(1, lambda=1)
# or simply:
dpois(2, lambda=1)

# Example 2 from class notes on the Poisson:
# X is Poisson with lambda=6.

# P(X >= 5) = 1 - P(X <= 4):
1 - ppois(4, lambda=6)

# P(X <= 9) - P(X<=6):
ppois(9, lambda=6) - ppois(6, lambda=6)

# Another example:

#  Suppose our random variable X is Poission with lambda = 12.33. 

#  Let's answer the following questions 
#  1. What is the probability of 15 or fewer occurrences?  P(X <= 15) 
ppois(15, lambda=12.33)

#  2. What is the probability of EXACTLY 6 successes? P(X = 6) 
ppois(6, lambda=12.33) - ppois(5, lambda=12.33)
# or simply:
dpois(6, lambda=12.33)

#  3. What is the probability of more than 13 successes? P(X > 13) 
1 - ppois(13, lambda=12.33)

#  4. What is the probability of 13 or more successes? P(X >= 13) 
1 - ppois(12, lambda=12.33)

#  5. What is the probability of 8, 9, or 10 successes? P(8 <= X <= 10) 
ppois(10, lambda=12.33) - ppois(7, lambda=12.33)

#  This R program will give us the answers to all these questions in an instant.