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• Hello again!

• In this lecture we are going to discuss the Bernoulli distribution.

• Before we begin, we useBernto define a Bernoulli distribution, followed by the

• probability of our preferred outcome in parenthesis.

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

• distribution with a probability of success equal to “p””.

• Okay!

• We need to describe what types of events follow a Bernoulli distribution.

• Any event where we have only 1 trial and two possible outcomes follows such a distribution.

• These may include a coin flip, a single True or False quiz question, or deciding whether

• to vote for the Democratic or Republican parties in the US elections.

• Usually, when dealing with a Bernoulli Distribution, we either have the probabilities of either

• event occurring, or have past data indicating some experimental probability.

• In either case, the graph of a Bernoulli distribution is simple.

• It consists of 2 bars, one for each of the possible outcomes.

• One bar would rise up to its associated probability of “p”, and the other one would only reach

• “1 minus p”.

• For Bernoulli Distributions we often have to assign which outcome is 0, and which outcome

• is 1.

• After doing so, we can calculate the expected value.

• Have in mind that depending on how we assign the 0 and the 1, our expected value will be

• equal to either “p” or “1 minus p”.

• We usually denote the higher probability with “p”, and the lower one with “1 minus

• p”.

• Furthermore, conventionally we also assign a value of 1 to the event with probability

• equal to “p”.

• That way, the expected value expresses the likelihood of the favoured event.

• Since we only have 1 trial and a favoured event, we expect that outcome to occur.

• By plugging in “p” and “1 minus p” into the variance formula, we get that the

• variance of Bernoulli events would always equal “p, times 1 minus p”.

• That is true, regardless of what the expected value is.

• Here’s the first instance where we observe how elegant the characteristics of some distributions

• are.

• Once again, we can calculate the variance and standard deviation using the formulas

• we defined earlier, but they bring us little value.

• For example, consider flipping an unfair coin.

• This coin is calledunfairbecause its weight is spread disproportionately, and it

• gets tails 60% of the time.

• We assign the outcome of tails to be 1, and p to equal 0.6.

• Therefore, the expected value would be “p”, or 0.6.

• If we plug in this result into the variance formula, we would get a variance of 0.6, times

• 0.4, or 0.24.

• Great job, everybody!

Hello again!

B1 中級

# ベルヌーイ配布 (Bernoulli Distribution)

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林宜悉 に公開 2021 年 01 月 14 日