
Schematically:


Example: sorting algorithm

Example: post office (when the next cunstomer arrive is a probabilitic event)
1. Ҏ[ one customer arrives in the next time interval Δt ] = &lambda×Δt + o(Δt) ........ (1) 2. Ҏ[ no customer arrives in the next time interval Δt ] = 1  &lambda×Δt + o(Δt) ........ (2) 3. Ҏ[ ≥ 2 customers arrive in the next time interval Δt ] = o(Δt) ........ (3) 4. The arrivals in nonoverlapping time intervals are (probabilistically) independent 
Note:

p(k) = Ҏ( k arrivals in an interval T ) 
Graphically:
k arrival events     V V V V <> T sec 

E[x] = ∑ _{(all values k) } k Ҏ[k] (λT)^{k} = ∑ _{(k = 0 .. ∞) } k ×  e^{λT} k! (λT)^{k} = ∑ _{(k = 1 .. ∞) }  e^{λT} (k1)!

E[x] = λT 
The random variable x represents the number of arrivals in a time interval of duration T (See: click here )

In other words:



Ҏ[ y > t ] = Ҏ[ no arrivals in interval (0..t) ] (λt)^{0} =  e^{λt} 0! = e^{λt} 
d Q(t) p(t) =  ..... (from Probability theory) dt 


Ҏ[ no arrival occurs within next t sec  no arrival for u sec ] = Ҏ[ no arrival occurs within next t sec ] 
Proof:
Ҏ[ no arrival occurs within next t sec  no arrival for u sec ] = Ҏ[ no arrival occurs within next t sec and no arrival for u sec ] =  (def of cond probability) Ҏ[ no arrival for u sec ] 