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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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# Probability: Binomial Distribution

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林宜悉 posted on 2020/03/09
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