HW2 - Comments on Poisson

Fundamentals of Data Science

The Poisson Distribution

The Poisson distribution measures the probability of \(k\) events occurring in a given time interval, assuming:

• the interarrival times are independent of one another
• the mean number of arrivals in any interval is a constant \(\lambda\).

pdata <- tibble(
    x = seq(0, 20),
)
plts <- list()
i <- 1

for (n in seq(1, 10, 2)) {
    cname <- paste0("y", n)
    pdata <- pdata |> mutate({{ cname }} := dpois(`x`, n))
}

p1 <- ggplot(data = pdata, aes(x = x)) +
    geom_col(aes(y = y1)) +
    ggtitle("mean=1")
p3 <- ggplot(data = pdata, aes(x = x)) +
    geom_col(aes(y = y3)) +
    ggtitle("mean=3")
p5 <- ggplot(data = pdata, aes(x = x)) +
    geom_col(aes(y = y5)) +
    ggtitle("mean=5")
p7 <- ggplot(data = pdata, aes(x = x)) +
    geom_col(aes(y = y7)) +
    ggtitle("mean=7")
p9 <- ggplot(data = pdata, aes(x = x)) +
    geom_col(aes(y = y9)) +
    ggtitle("mean=9")
grid.arrange(p1, p3, p5, p7, p9)

The Exponential Distribution

The exponential distribution is a continuous probability distribution given by an exponential function with a fixed rate \(\lambda\).

x <- seq(0, 3, .01)
y <- dexp(x, 1)
ggplot() +
    geom_line(aes(x = x, y = y)) +
    ggtitle("Exponential Distribution with parameter 1")

Poisson Process

In a “Poisson Process”, events occur randomly in time. The interval between two consecutive events is chosen (independently) from an exponential distribution.

t <- rexp(100, 1)
arrivals <- cumsum(t)
ggplot() +
    geom_col(aes(x = arrivals, y = 1), width = .015, color = "black") +
    ggtitle("Arrival Times in a Poisson Process")

sum(arrivals < 25)

[1] 32

sum(arrivals < 50)

[1] 50

sum(arrivals > 25 & arrivals < 50)

[1] 18