Let T be the time for the delivery of a particular item, R be the time required for process the order, and S be the shipping time. Then, the minimum amount of time for the delivery of the item is T = R + S. In general, both R and S are random variables; hence T is also a random variable. For the sake of simplicity, we shall consider only the case of instant processing for an order, i.e. R = 0. Then, the delivery time T equals the shipping time S.
Since T is a random variable, the chance that an item will be delivered on day t is represented by the probability p(t). Then, the probability that the item will be delivered on or before t day is given by