From the previous two results, \[ \P(X_1 = x_1, X_2 = x_2, \ldots, X_n = x_n) = \frac{a^{(c,k)} b^{(c, n - k)}}{(a + b)^{(c,n)}} =\frac{[a^{(c,k)} / c^k] [b^{(c, n - k)} / c^{n-k}]}{(a + b)^{(c,n)} / ...From the previous two results, \[ \P(X_1 = x_1, X_2 = x_2, \ldots, X_n = x_n) = \frac{a^{(c,k)} b^{(c, n - k)}}{(a + b)^{(c,n)}} =\frac{[a^{(c,k)} / c^k] [b^{(c, n - k)} / c^{n-k}]}{(a + b)^{(c,n)} / c^n} = \frac{(a /c)^{[k]} (b / c)^{[n-k]}}{(a / c + b / c)^{[n]}} \] and this is the corresponding finite dimensional distribution of the beta-Bernoulli distribution with parameters \( a / c \) and \( (b / c) \).
Next, let \( \bs{Y} = \{Y_n: n \in \N\} \) denote the partial sum process associated with \( \bs{X} \), so that once again, \[ Y_n = \sum_{i=1}^n X_i, \quad n \in \N \] Of course \( Y_n \) is the numb...Next, let \( \bs{Y} = \{Y_n: n \in \N\} \) denote the partial sum process associated with \( \bs{X} \), so that once again, \[ Y_n = \sum_{i=1}^n X_i, \quad n \in \N \] Of course \( Y_n \) is the number of success in the first \( n \) trials and has the beta-binomial distribution defined by \[ \P(Y_n = k) = \binom{n}{k} \frac{a^{[k]} b^{[n-k]}}{(a + b)^{[n]}}, \quad k \in \{0, 1, \ldots, n\} \] Now let \[ Z_n = \frac{a + Y_n}{a + b + n}, \quad n \in \N\] This variable also arises naturally.
