n: #trials $\quad$ p:P(success) $\quad$ X:# success in n_trial
If each trial is independent , then distribution is Bin(n,p) . Here we are looking what $ i^th $ trial results in ? We can also answer questions like P(X = k)
Up until now , things were simple. If I plot at what time trial took plce from some start time , then spacing between 2 trials can be totally random on time axis (like taking 1 , resting for hour and then taking 2 back to back and so on ...)
Now , I can enquire about P(#success = k in time $t_0$). Here we can divide our time frame into finer and finer intervals to avoid overlap( >1 trials in an interval). $''This\quad description\quad leads\quad to\quad poisson \quad distribution ,\quad if\quad I\quad understood\quad correctly ''$
But I have certain arguments::: In certain time interval, it is possible that no trial took place, leave away getting success. According to me-->$$ P(success \quad in \quad i^th slot) = P(trial\quad in\quad i^th\quad slot ,\quad success\quad in\quad trial) = P(trial\quad in\quad i^th\quad slot )*p$$
$$ P(no\quad success \quad in \quad i^th slot) = P(no\quad trial\quad in\quad i^th\quad slot) + P(trial\quad in\quad i^th\quad slot , \quad no\quad success\quad in\quad trial)$$
Though both $ P(success \quad in \quad i^th slot)\quad and\quad P(no\quad success \quad in \quad i^th slot)\quad $ add to 1 but $ \quad P(trial\quad in\quad i^th\quad slot) \quad $ is HGeom distribution (not independent of previou (i-1) slots)
$$Please\quad help ,\quad also\quad shed\quad some\quad light\quad on\quad rate\quad parameter\quad (lambda) \quad in \quad Poisson\quad distribution $$
Thanks in advance