Derivation of exponential probability distribution of a process duration assuming constant probability rate of process termination

Suppose a process has begun and its probability of termination in a time interval dt is ldt. The probability is assumed independent on time. The probability of a process to survive time t+dt is equal to the probability of a process to survive time t and to not terminate in a next time interval dt. Thus:

P(t + dt) = P(t)(1 - ldt) dP(t) = -P(t)ldt dP(t)/dt = -P(t)l Solution to this differential equation is: P(t) = P0e-lt Where P0 is a normalization constant: P0 = 1/∫e-ltdt We finally get: P(t) = le-lt which is the exponential probability density function.