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  • In essence, Binomial events are a sequence of identical Bernoulli events.

  • Before we get into the difference and similarities between these two distributions, let us examine

  • the proper notation for a Binomial Distribution.

  • We use the letter “B” to express a Binomial distribution, followed by the number of trials

  • and the probability of success in each one.

  • Therefore, we read the following statement asVariable “X” follows a Binomial

  • distribution with 10 trials and a likelihood of success of 0.6 on each individual trial”.

  • Additionally, we can express a Bernoulli distribution as a Binomial distribution with a single trial.

  • Alright!

  • To better understand the differences between the two types of events, suppose the following

  • scenario.

  • You go to class and your professor gives the class a surprise pop-quiz, which you have

  • not prepared for.

  • Luckily for you, the quiz consists of 10 true or false problems.

  • In this case, guessing a single true or false question is a Bernoulli event, but guessing

  • the entire quiz is a Binomial Event.

  • Alright!

  • Let’s go back to the quiz example we just mentioned.

  • In it, the expected value of the Bernoulli distribution suggests which outcome we expect

  • for a single trial.

  • Now, the expected value of the Binomial distribution would suggest the number of times we expect

  • to get a specific outcome.

  • Great!

  • Now, the graph of the binomial distribution represents the likelihood of attaining our

  • desired outcome a specific number of times.

  • If we run n trials, our graph would consist “n + 1”-many bars - one for each unique

  • value from 0 to n.

  • For instance, we could be flipping the same unfair coin we had from last lecture.

  • If we toss it twice, we need bars for the three different outcomes - zero, one or two

  • tails.

  • Fantastic!

  • If we wish to find the associated likelihood of getting a given outcome a precise number

  • of times over the course of n trials, we need to introduce the probability function of the

  • Binomial distribution.

  • For starters, each individual trial is a Bernoulli trial, so we express the probability of getting

  • our desired outcome as “p” and the likelihood of the other one as “1 minus p”.

  • In order to get our favoured outcome exactly y-many times over the n trials, we also need

  • to get the alternative outcome “n minus y”-many times.

  • If we don’t account for this, we would be estimating the likelihood of getting our desired

  • outcome at least y-many times.

  • Additionally, more than one way to reach our desired outcome could exist.

  • To account for this, we need to find the number of scenarios in which “y” out of the “n”-many

  • outcomes would be favourable.

  • But these are actually thecombinationswe already know!

  • For instance, If we wish to find out the number of ways in which 4 out of the 6 trials can

  • be successful, it is the same as picking 4 elements out of a sample space of 6.

  • Now you see why combinatorics are a fundamental part of probability!

  • Thus, we need to find the number of combinations in which “y” out of the “n” outcomes

  • would be favourable.

  • For instance, there are 3 different ways to get tails exactly twice in 3 coin flips.

  • Therefore, the probability function for a Binomial Distribution is the product of the

  • number of combinations of picking y-many elements out of n, times “p” to the power of y,

  • times “1 - p” to the power of “n minus y”.

  • Great!

  • To see this in action, let us look at an example.

  • Imagine you bought a single stock of General Motors.

  • Historically, you know there is a 60% chance the price of your stock will go up on any

  • given day, and a 40% chance it will drop.

  • By the price going up, we mean that the closing price is higher than the opening price.

  • With the probability distribution function, you can calculate the likelihood of the stock

  • price increasing 3 times during the 5-work-day week.

  • If we wish to use the probability distribution formula, we need to plug in 3 for “y”,

  • 5 for “n” and 0.6 for “p”.

  • After plugging in we get: “number of different possible combinations of picking 3 elements

  • out of 5, times 0.6 to the power of 3, times 0.4 to the power of 2”.

  • This is equivalent to 10, times 0.216, times 0.16, or 0.3456.

  • Thus, we have a 34.56% of getting exactly 3 increases over the course of a work week.

  • The big advantage of recognizing the distribution is that you can simply use these formulas

  • and plug-in the information you already have!

  • Alright!

  • Now that we know the probability function, we can move on to the expected value.

  • By definition, the expected value equals the sum of all values in the sample space, multiplied

  • by their respective probabilities.

  • The expected value formula for a Binomial event equals the probability of success for

  • a given value, multiplied by the number of trials we carry out.

  • After computing the expected value, we can finally calculate the variables.

  • After computing the expected value, we can finally calculate the variance.

  • We do so by applying the short formula we learned earlier:

  • Variance of Y equals the expected value of Y squared, minus the expected value of

  • Y, squared.”

  • After some simplifications, this results in “n, times p, times 1 minus p”.

  • If we plug in the values from our stock market example, that gives us a variance of 5, times

  • 0.6, times 0.4, or 1.2.

  • This would give us a standard deviation of approximately 1.1.

  • Knowing the expected value and the standard deviation allows us to make more accurate

  • future forecasts.

In essence, Binomial events are a sequence of identical Bernoulli events.

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B1 distribution probability outcome likelihood bernoulli expected

Probability: Binomial Distribution

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    林宜悉 posted on 2020/03/09
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