def conditional_expected_number_of_purchases_up_to_time(self, t, frequency, recency, T):
        """
        Conditional expected number of purchases up to time.

        Calculate the expected number of repeat purchases up to time t for a
        randomly choose individual from the population, given they have
        purchase history (frequency, recency, T)

        Parameters
        ----------
        t: array_like
            times to calculate the expectation for.
        frequency: array_like
            historical frequency of customer.
        recency: array_like
            historical recency of customer.
        T: array_like
            age of the customer.

        Returns
        -------
        array_like

        """
        x = frequency
        r, alpha, a, b = self._unload_params("r", "alpha", "a", "b")

        _a = r + x
        _b = b + x
        _c = a + b + x - 1
        _z = t / (alpha + T + t)
        ln_hyp_term = np.log(hyp2f1(_a, _b, _c, _z))

        # if the value is inf, we are using a different but equivalent
        # formula to compute the function evaluation.
        ln_hyp_term_alt = np.log(hyp2f1(_c - _a, _c - _b, _c, _z)) + (_c - _a - _b) * np.log(1 - _z)
        ln_hyp_term = np.where(np.isinf(ln_hyp_term), ln_hyp_term_alt, ln_hyp_term)
        first_term = (a + b + x - 1) / (a - 1)
        second_term = 1 - np.exp(ln_hyp_term + (r + x) * np.log((alpha + T) / (alpha + t + T)))

        numerator = first_term * second_term
        denominator = 1 + (x > 0) * (a / (b + x - 1)) * ((alpha + T) / (alpha + recency)) ** (r + x)

        return numerator / denominator