So I am a little confused about the function for the number of censored lives aged x at time t and how exposure to risk is estimated using the census data and the trapezoidal rule. Since we assume the function continuous with time, does that mean it increases and decreases according to deaths, withdrawals and new entries? If so, is it possible to get a probability qx greater than 1, if we experience a large amount of deaths and use the trapezoidal rule? (Under the binomial model)

I think this is theoretically possible, say if \(P_x(t) = 100\), \(P_x(t+1) = 1000\) and \(d_x = 600\). But this would be very peculiar and, if it happened, you may have bigger problems!