Probabilistic forecasting

My question relates to probabilistic forecasting. How does one actually go about computing a forecast?

Lets say I have some data that can be modelled by a specific distribution, and the values of the coefficients etc. are known. Therefore, I know the mean of this distribution gives me the expected value, but is this value the forecast? How would the forecast progress?

For example, with ARMA forecasts, the forecasted values are computed using a recursive formula which is updated at each time step with the new value. I don't see how this applies to a probabilistic forecast.

Is it perhaps random (the term might be stochastic)? As in, for each time you want to forecast, you randomly generate a value from the distribution?

Thanks.

With a probabilistic forecast, you use your glm with covariates at times, say, $t-1$ and $t$ for forecasting the distribution at time $t+1$. Now, this forecasted distribution can be used to find the mean, median, confidence intervals etc. whatever is useful for disseminating your forecast. For example, if you are forecasting a rainfall distribution you can use this to say what the expected rainfall value at time $t+1$ is, or you can give a probability of it exceeding some threshold based on the tails of the distribution.
Then when new data becomes available i.e. at time $t+1$ you can then use these observations to forecast the distribution for time $t+2$. So the procedure is still recursive as it is for ARMA, the difference is probabilistic forecasts recursively forecast a distribution as opposed to just the mean.