Chapter 33: Gamma Random Variables

Week 10
Author

Meike Niederhausen and Nicky Wakim

Published

November 29, 2023

Learning Objectives

  1. Identify the variable and the parameters in a story, and state in English what the variable and its parameters mean.

Properties of gamma RVs

  • Scenario: Modeling the time until the \(r^{th}\) event.
  • Continuous analog to the Negative Binomial distribution
  • Shorthand: \(X \sim \text{Gamma}(r, \lambda)\)

\[ f_X(x) = \dfrac{\lambda^r}{\Gamma(r)}x^{r-1} e^{-\lambda x}\text{ for } x>0, \lambda>0, \Gamma(r) = (r-1)! \]

\[ F_X(x) = \left\{ \begin{array}{ll} 0 & \quad x<0 \quad \\ 1 - e^{-\lambda x}\displaystyle\sum_{j=0}^{r-1}\dfrac{(\lambda x)^j}{j!} & \quad x\geq0 \\ \end{array} \right. \]

\[\text{E}(X) = \dfrac{r}{\lambda}\text{, }\text{ Var}(X) = \dfrac{r}{\lambda^2}\]

Common to see \(\alpha = r\) and \(\beta = \lambda\)

Identifying gamma RV from word problems

  • Gamma distribution with \(r=1\) is same as exponential

    • Just like Negative Binomial with \(r=1\) is same as the geometric distribution

  • Similar to exponential

    • Look for time between or until events/successes
      • BUT now we are measuring time until more than 1 success
    • Look for a rate of the events over time period

Helpful R code

Let’s say we’re sitting at the bus stop, measuring the time until 4 buses arrive. We know the bus comes every 10 minutes on average.

  • If we want to know the probability that the 4 buses arrive in the next 50 minutes:

    pgamma(q = 50, rate = 1/10, shape = 4)
    [1] 0.7349741
    pgamma(q = 50, scale = 10, shape = 4)
    [1] 0.7349741

  • If we want to know the time, say \(t\), where the probability of the 4 buses arriving at \(t\) or earlier is 0.35:

    qgamma(p = 0.35, rate = 1/10, shape = 4)
    [1] 29.87645

  • If we want to know the probability that the 4 buses arrives between 30 and 50 minutes:

    pgamma(q = 50, scale = 10, shape = 4) - pgamma(q = 30, scale = 10, shape = 4)
    [1] 0.382206

  • If we want to sample 20 arrival times for the 4 buses:

    rgamma(n = 20, scale = 10, shape = 4)
     [1] 23.62559 67.87353 66.09070 37.22996 26.01786 69.19651 33.56302 41.93588
 [9] 18.28835 76.26211 74.34490 51.98992 15.32024 50.57118 73.52642 36.96576
[17] 66.41893 17.71069 30.96120 21.86717

Remarks

  • The parameter \(r\) in a Gamma(\(r\),\(\lambda\)) distribution does NOT need to be a positive integer

    • \(r\) is usually a positive integer

  • When \(r\) is a positive integer, the distribution is sometimes called an Erlang(\(r\),\(\lambda\)) distribution

     

     

  • When \(r\) is any positive real number, we have a general gamma distribution that is usually instead parameterized by \(\alpha>0\) and \(\beta>0\), where:

    • \(\alpha = \text{shape parameter}\) : same as \(k\), the total number of events we must witness

      • In R code example: 4 buses to wait for

    • \(\beta = \text{scale parameter}\) : same as \(\lambda\), the rate parameter

      • In R code example: 1 bus per 10 minutes (1/10)

Sending money orders

Example 1

On average, someone sends a money order once per 15 minutes. What is the probability someone sends 10 money orders in less than 3 hours?

Additional Resource