Bernoulli

What It Is

This may be one of the most intuitive distributions. The Bernoulli distribution models the success/failure of a series of singular outcomes represented by a single parameter:

p probability of success

Classically, this type of distribution is easily described by successive coin flips or trials, where an outcome of heads would be represented by a value of 1, and likewise, tails a value of 0. Assuming a balanced or unbiased coin, there is equal probability of either outcome, p = 0.50.

The above plot shows the results of 1000 trials. The expected number of occurrences for each outcome is represented by the red points. The bars represent the observed outcomes of the sampling, filling up as replications are completed.

📑 Note: Due to the randomness of the Sample data, the number of observed outcomes does not exactly equal the expected number of observed successes and failures but is comparable.

Variants

The following plots illustrate the expected and random observations for varying the value of p, resulting in different shapes.

p = 0.85

With values of p > 0.5, there is an obvious bias towards successful outcomes.

p = 1.0

Where p = 1.0 there is only one possible outcome; there is no need to model the distribution.

In Real Life (IRL)

How would you use a Bernoulli distribution IRL? With caution! This distribution should be used for simple two-state outcomes. If more outcomes exist, then it would be more appropriate to use one of the other distributions that will be explored in this series.

Example A

You are the quality assurance director of a mid-tier paper company. You ensure that the received pallets of 8.5x11 65lb ultra white card stock meets the strict quality standards of your company. Your boss wants to know the rate of undamaged received pallets. How will you report this?

For each shipment, you inspect each pallet. If the pallet is received without damage, you score it as "Pass" and represent it with a value of one (1), else it is a "Fail" and assigned a value of zero (0).

Upon completion, you have the total number of passing/failing pallets. You set p = (number of Pass) / (number of inspected pallets) and make a fancy chart to present your results.

Example B

You are a professor at an Ivy League research university in Ithaca, New York. As provide the utmost level of instruction for your History 291 class you carefully monitor the rate of students passing or failing your class. The dean wants to see your how often students fail your course.

At the end of each semester, you score categorize student grades as "Pass" or "Fail", representing these scores with a value of one (1), and zero (0), respectively.

Upon completion, you have the total number of passing/failing students. You set p = (number of Pass) / (number of inspected pallets) and make a fancy chart to present your results. You report to the dean that your fail rate is (1 - p) * 100.

Again, having only one parameter, generating this type of model is not difficult as it is just a proportion of two outcomes. Any two-state outcome can be evaluated a significant number of times to determine a consistent value of p.

Conclusion

This is a basic probability distribution. However, determining the appropriate value to model a component of your system would require some insight and ensuring that the appropriate number of observations are recorded.

Now that you have been introduced to the Bernoulli distribution, know that it is a special case of the distribution featured in the next article of this series. Continue to read more about the binomial distribution.

YHWH, your yes/no, pass/fail, is sufficient. Help us to accept your direction in our life.

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