Halloween and the rare events
Tales of ghosts, sweets, and photons
Tales of spirits who return from the dead are present in the folklore of many cultures. Halloween gives out the perfect context to introduce a probabilistic model, which entails an element of randomness. This word does not have to be understood as its typical mathematical definition. It should be understood as a probabilistic model that can display random behavior of certain events occurring in intervals of time (or space). Ghostly apparitions are rare events occurring independently in time. Do not co-occur at the same instant. And the number of spectra manifestations (events) in a unit time interval is likely equally possible to happen in the next unit interval at the same rate.
The Vanishing hitchhiker!
Have you heard of the Vanishing hitchhiker urban legend? If you like a fright, consider finding a spot where reports have been made of this phenomenon. Drive up to that legendary spot close to your town where local folklore talks about a hitchhiker who appears and suddenly fades. The Poisson distribution can describe this situation since hitchhiker ghost appearances are independent, not simultaneous, and with an average of λ ≥ 0 times per hour (or another time unit, if I may say that). The probability P(n) of the Vanishing hitchhiker appearing in any given hour n is:
The Poisson distribution is a discrete probability first derived in 1837 by the French mathematician and engineer Siméon Denis Poisson. Gives the probability that a finite interval λ (time or space) contains exactly n > 0 events. Where e is the Euler’s number and n! is the factorial of n (the product of all integers from 1 to n).
On a 3-item ranking, there are 3! = 3·2·1 = 6 different ways of arranging them into an ordered list (ranking).
Considering that possibility that the hitchhiker ghost (n = 1) can appear once per hour (λ = 1), P(1) = 1/e ~0.3678. Then the probability of seeing it is just over 36%. That is not bad!
The Poisson distribution has been used as a likelihood function to estimate the epidemic reproduction number of COVID-19 in real-time.
Trick-or-treat!
Are you planning to stay at home this Halloween? Dress up, decorate your home sinister and gloomy and wait for the kids from your neighborhood to knock on your door while they scream the Halloween dilemma: Trick-or-treat!!!!. Well, you must say, Trick!
What is the probability of 5 kids knocking at your door between 6 pm and midnight? When the events can occur over time, continuously and independently, that is called a Poisson process. Therefore, adding the time period T = 6 hours (considering λ = 1 as a reasonable value) to the Poisson Distribution:
It is 16%! Prepare some sweets! And that’s just for starters.
The Poisson process has been used to estimate the number of advanced intelligent civilizations that can communicate in the Milky Way disk as a way to introduce a temporal aspect to the Drake equation formula.
Photon Noise
Are you going to take pictures at your Halloween party? Be sure that you have the right light. One source of noise in optics is the photon noise, also called Poisson noise due to the Poissonian nature of counting photons of a CCD. A CCD (Coupled-Charged Device) can be seen as an electronic eye that collects light and converts it into an electrical signal. CCD's main application is digital imaging (e.g., photography, digital video) and when a high-quality image is required (medicine, astrophotography, etc.).
Photons are the particles of light intuited by Einstein.
This type of noise is not something intrinsic to the device. The larger the number of photons gathered, the lesser the noise.
Don’t confuse this type of noise with the “snow” when an analog TV had no channel to display.
Now you have the tools to determine the chances you have to find an analog TV in your neighborhood (space-oriented Poisson application) before Halloween.
