Probability of Exponential Decay

Given the exponential decay operation

x (t) = x (0) e^{- α t}

$α$ is the rate of decay
$α (d t)$ is the probability of decay in the next infinitessimal time interval $d t$ .

P (exisiting at time t) = \frac{x (t)}{x (0)} = e^{- α t}

let's find the probability that one particle did not decay at any time between time 0 and time $t$ .

If we consider infinitely many small intervals $d t$ , where $n \cdot d t = t$ (thus $n$ is close to infinity)

P (not decaying in d t) = 1 - α d t

So, the probability of the particle not decaying during any of these tiny intervals:

P (exisiting at time t) = (1 - α d t)^{n}

Taking the logarithm:

l n (P (exisiting at time t)) = n \cdot l n (1 - α d t)

Consider the plots of $y = l n (x)$ and $y = x - 1$ :

We can see that $l n (x) \approx x - 1$ when $x \approx 1$ . Since we were considering $x = 1 - α d t$ , this means that when

At this point, $- n α d t$

Conclusion

Exponential distribution: The probability of dying between $t$ and $t + d t$ is given by:

P (decay between t and t + d t) = e^{- α t} α d t

We call the important part of this the probability density function. This is a kind of Continuous Distributions.

f (t) = e^{- α t} α