5.3 The geometric distribution (discrete)

Deriving the geometric distribution is best done using an example.

Example 5.2 (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 5.3 (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 $0 < p < 1$.