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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!

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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
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