Questions about modelling using the exponential distribution (Probability - Jim Pitman self study)

Question

I've been reading the chapter 4.2 from Jim Pitman's Probability book. The chapter talks about the exponential distribution and it has two examples of modelling using the exponential distribution and I need some clarifications.

First question - reliability

The first one is about modelling the lifetime $T$ of an electrical component. He is saying that the exponential distribution is a good model for it and he is making the following assumptions:

• The component does not wear out gradually rather it stops working suddenly and unpredictably.
• No matter how long the component has been in use, the chance that it survives a further time interval of length $\Delta$ is always the same.

Then he concludes that the probability must be $e^{-\lambda \Delta}$.

Can someone please check if my understanding of this and my derivation of the exponential distribution is correct:

The first point made is saying that there is no ageing effect in the lifetime of the component and that there is randomness of the failure time so a continuous probability distribution without ageing effect is a good model for the lifetime of the component.

The second is saying that $P(T > t + \Delta | T>t)$ is independent of $t$. Then on one hand we have that $$P(T > t + \Delta | T>t) = \frac{P(T> t + \Delta)}{P(T>t)}$$ and on the other from the independence assumption for $t=0$ we get that $$P(T > t + \Delta | T>t) = P(T > 0 + \Delta | T>0) = P(T>\Delta).$$

Equating the above gives us that $T$ has the memoryless property and it is known that the only memoryless continuous probability distribution is the exponential distribution.

Second question - radioactive decay

The assumptions here are: $T$ is the random lifetime, or time until decay of an atom and that the distribution of $T$ has the memoryless property so $T$ has the exponential distribution $exp(\lambda)$ for some $\lambda >0$.

All good until here but I do not understand the following:

Probabilities have a clear interpretation due to the large number of atoms involved. Assume that the number of atoms is a large number $N$ and that they decay independently of each other. Then by the law of large numbers the proportion of these $N$ atoms that survives up to time $t$ is bound to be close to $e^{-\lambda t}$, the survival probability of each individual atom.

Can someone please explain this part?

Thanks

Answer 1

Then he concludes that the probability must be $e^{-\lambda \Delta}$.

I take it you mean $P(T>\Delta)=e^{-\lambda \Delta}$

The second is saying that $P(T > t + \Delta | T>t)$ is independent of $t$.

yes. This is an important point.$$P(T > t + \Delta | T>t)=\frac{P(T > t + \Delta )}{P(T>t)}=\frac{e^{-\lambda(t+\Delta)}}{e^{-\lambda t}}=e^{-\lambda\Delta}$$

All good until here but I do not understand the following: $\dots\dots$

Let $X_i$ be the indicator random variable, for the event "$i^{th}$ atom survives after time $t$".

Hence $X_i=1$, if $i^{th}$ atom survives after time $t$. And $X_i=0$, if $i^{th}$ atom decays in time period $[0,t]$

Then the proportion of atoms that survive is $$\bar{X}=\frac{\sum_{i=1}^N X_i}{N}$$

By LLN,

$\bar{X}\xrightarrow[n \to \infty]{}E[X_i]=P(X_i=1)=P(T>t)=e^{-\lambda t}$