The Bernoulli Distribution. One of the first stats concepts that I learned in middle school. Interested in seeing what else this mathematician did, I looked him up and was surprised to see how many concepts were named after him. The Bernoulli Effect, the Bernoulli Series, Bernoulli Polynomials. I remember thinking, "man this guy must have been really smart or had nothing better to do". However, everything made more sense a couple years later when I found out they a whole family of geniuses and each had come up with their own mark in math history. It was probably the only way to not be the loser of the family.
This was Johann Bernoulli
What is the Bernoulli Function?
Anyways Jacob Bernoulli was the one who coined this distribution. The idea behind it is that in any event, you will either get a success (1) or a failure (0), and the probability of getting the success is p, making the probability of failure (1-p). Say you were trying to pick an ace out of a randomly shuffled deck. Since there are four aces in a deck with fifty two cards, the probability of success or p is 1/13, which means the probability of failure is 12/13. When repeated an n number of times the expected value or chance that you will succeed n number of times in picking an ace each time decreases exponentially. These probabilities can be graphed, creating the Bernoulli Distribution.
Formula for calculating the Bernoulli function
Formula for calculating expected value and variance
Where is it used?
This distribution can be used in any and every statistical modeling as long as it meets four criteria.
1) It has only two outcomes -- Success or Failure.
(Some event either can or cannot happen)
2) Each trial must be independent from one another.
(The success or failure in trial one cannot affect the success
or failure in outcome two)
3) Probability of success and failure stays the same throughout
each trial.
4) Number of trials are fixed.
So going back to our picking aces example. We can use the Bernoulli Distribution to try and guess how many times we will succeed as long as we put the card I drew back into the deck before I run the experiment again. If I don't then criteria 2 and 3 will not apply so I cannot use the distribution to calculate my success over n number of times.
Pictures and Information gotten from:
https://probabilityformula.org/bernoulli-trials.html
https://towardsdatascience.com/understanding-bernoulli-and-binomial-distributions-a1eef4e0da8f
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