Set Up

mean_calls <- 15

Questions

What is the probability that she will hear more than 8 calls during any given session?

p_more_than_8 <- 1 - ppois(8, mean_calls)
cat("P(X > 8):", round(p_more_than_8, 4), "\n")
## P(X > 8): 0.9626

What is the probability that she will hear no calls in a session?

p_zero_calls <- dpois(0, mean_calls)
cat("P(X = 0):", round(p_zero_calls, 4), "\n")
## P(X = 0): 0

What is the probability that she will hear exactly 3 calls in a session?

p_three_calls <- dpois(3, mean_calls)
cat("P(X = 3):", round(p_three_calls, 4), "\n")
## P(X = 3): 2e-04

Plot the relevant Poisson mass function over the values in range 0 ≤ x ≤ 30.

x_values <- 0:30
plot_mass_values <- dpois(x_values, mean_calls)
plot(x_values, plot_mass_values, type = "h", lwd = 2, col = "blue",
main = "Poisson Probability Mass Function (mean_calls = 15)",
xlab = "Number of Calls", ylab = "Probability")
points(x_values, plot_mass_values, pch = 16, col = "black")

#Simulate 104 results from this distribution (i.e., 2 years of Saturday monitoring sessions).

set.seed(42)
two_years_data <- rpois(104, mean_calls)

#Plot the simulated results using hist() and use xlim() to set the horizontal limits to be from 0 to 30. How does your histogram compare to the shape of the probability mass function you plotted above?

hist(two_years_data, breaks = 30, freq = FALSE, col = "red",
main = "Simulated Poisson Distribution (104 sessions)",
xlab = "Number of Calls", xlim = c(0, 30))
lines(x_values, plot_mass_values, type = "b", col = "black", lwd = 2, pch = 16)