2.8 Gamma

If \(X\) is the interval until the \(\alpha^{th}\) successful event when the average interval is \(\theta\), then \(X\) is a random variable with a gamma distribution \(X \sim \Gamma(\alpha, \theta)\). The probability of an interval of \(X = x\) is

\[f(x; \alpha, \theta) = \frac{1}{\Gamma(\alpha)\theta^\alpha}x^{\alpha-1}e^{-x/\theta}.\]

where \(\Gamma(\alpha) = (1 - \alpha)!\) with \(E(X) = \alpha \theta\) and \(Var(X) = \alpha \theta^2\).

Examples

On average, someone sends a money order once per 15 minutes (\(\theta = .25\)). What is the probability someone sends \(\alpha = 10\) money orders in less than \(x = 3\) hours?*

theta = 0.25
alpha = 10
pgamma(q = 3, shape = alpha, scale = 0.25)

## [1] 0.76

data.frame(x = 0:1000 / 100, prob = pgamma(q = 0:1000 / 100, shape = alpha, scale = theta, lower.tail = TRUE)) %>%
  mutate(Interval = ifelse(x >= 0 & x <= 3, "0 to 3", "other")) %>%
ggplot(aes(x = x, y = prob, fill = Interval)) +
  geom_area(alpha = 0.9) +
  theme_mf() +
  scale_fill_mf() +
  labs(title = "X ~ Gam(alpha = 10, theta = .25)",
       subtitle = "Probability of 10 events in X hours when the mean time to an event is .25 hours.",
       x = "Interval (x)",
       y = "pgamma")