Memoryless Property Let \(X\) be an exponentially distributed random variable.
Life of electronic components without moving parts are often modeled as exponential random variables. This means that if a component is already \(r\) years old, the probability it will last for \(s\) more years is equal to the probability a new identical component will last for \(s\) years. Meaning there is no point buying a new one, you should wait till failure to replace it and use a different way to manage failures when they happen.
Radioactive Decay It can be proved (those who are interested can try your hand at it, or see me for a proof) that the exponential random variable is the only continuous random variable defined on \([0,\infty)\) that can have the memoryless property. This has a very important consequence in physics, one that you have already studied.
Radioactivity is a probabilistic, quantum mechanical phenomenon, wherein certain atoms, like C-14, rearrange themselves spontaneously to a lower energy configuration. One may posit such a memoryless property for radioactive decay: a C-14 atom at time 0, regardless of past, will rearrange itself to N-14 at a time \(t\) in the future with some probability. The assumption here is not on the probability, only that the probability is independent of the past from time 0. This is enough to ensure that the time to decay is modeled by an exponential random variable, and all one has to do then is to find the exponential parameter by observing the half life of C-14! Of course, this has been verified in practice to fantastic precision, and the harbinger of the atomic age.
Let \(R_1, \ldots R_k\) be \(k\) independent exponentially distributed random variables with parameters \(\lambda_1,\ldots, \lambda_k\). Show that
\[R = \min_{1\le i \le k} R_i\]is also an exponentially distributed random variable. What is its parameter?
Poisson and Exponentials Let \(X_1, X_2, \ldots\) be iid exponential random variables with parameter \(\lambda\). Think of a packet queue at an Internet hub, and suppose \(X_i\) is the time gap (in seconds) between the \(i-1\)th and \(i\)‘th packet. Show that the number of packets that arrive in \(t\) seconds is Poisson distributed with parameter \(\lambda t\). Therefore, when you model interarrival times as exponential, the number of arrivals in any time interval is Poisson.