I would take the first day you possibly want to war and the last day. Maybe you want the war to start sometime between day 14 and 42... so you set the probability density function to k=dayofround-14 and lambda = (lastday-firstday)/2 (so (42-14)/2 in this example).

And the reason I choose poisson instead of gaussian is that you can skew the odds of having the war to being higher shortly after an intended prep time if you prefer... so maybe you want it to be in week 4 or 5 but you want higher odds of week 4 then poisson is built for that (and in that case you'd want to set lambda to be equal to 3.5)