Consider a situation where products are produced sequentially. The events producing defective products are independent and identically distributed, and a defective product is produced with a probability of $\phi$ $(0 \leq \phi \leq 1)$. Suppose that the probability of producing a defective product follows the Beta distribution
$$\operatorname{Beta}_{\mathrm{a},\mathrm{b}}(x) = \frac{1}{\mathrm{B}(\mathrm{a},\mathrm{b})} x^{\mathrm{a}-1}(1-x)^{\mathrm{b}-1} \quad (0 \leq x \leq 1),$$
for real numbers $\mathrm{a}(>1)$ and $\mathrm{b}(>1)$. Note that the Beta function $\mathrm{B}(\mathrm{a},\mathrm{b})$ is defined as
$$\mathrm{B}(\mathrm{a},\mathrm{b}) = \int_0^1 t^{\mathrm{a}-1}(1-t)^{\mathrm{b}-1} \mathrm{d}t$$
By defining $v_i = 1$ if the $i$-th product is a defective product, and $v_i = 0$ if it is not defective, we get a series $\boldsymbol{v} = (v_1, \cdots, v_N)$, where the values can be 0 or 1. Let $N_d(\boldsymbol{v})$ be the number of observations with value of 1 in $\boldsymbol{v}$.
Suppose that $Q(\boldsymbol{v} \mid \phi)$ in Question II.2 is the occurrence probability obtained in Question II.1 and let $a = 2,\ b = 50$, obtain $Q_{2,50}(\boldsymbol{v})$.