An insurer sells an annual policy with a claim frequency with the following distribution:

[[math]] p_k = \binom{4}{k}q^k(1-q)^{4-k}. [[/math]]

The parameter [math]q[/math] is unknown but your prior beliefs indicate that it has the following density function on [0,1]:

[[math]] 12 \, q^2(1-q). [[/math]]

Determine the expected annual claim frequency.

1. 0.6
2. 2.3
3. 2.4
4. 3.25
5. 4