## 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 $$0<p<1$$.