5.3 The geometric distribution (discrete)

Deriving the geometric distribution is best done using an example.

Example 5.1 (Deriving the geometric distribution) Suppose again that we know that the probability of finding a cracked egg is \(p=0.1\). How many eggs do we need to inspect before finding the first cracked egg?

The probability that the first egg is the cracked egg is~\(p=0.1\). The probability that the second egg we inspect is the cracked egg is the probability that the first egg is not cracked, but the next one is; that is, \(0.9\times 0.1\).

Similarly, the probability that the third we inspect is the cracked egg is the probability that the first two eggs are not cracked, but the next one is; that is, \(0.9\times 0.9\times 0.1\).

Continuing, this leads to the PMF of a geometric distribution.

Definition 5.1 (Geometric distribution) The PMF for a geometric distribution \[ p_X(x) = p\times (1-p)^{x-1}\quad\text{for $x=1, 2, \dots$}, \] for \(0<p<1\).