Original subtitles

-: Hello, again.

In this lecture,

we are going to discuss

the Bernoulli Distribution.

Before we begin,

we use Bern

to define

a Bernoulli Distribution,

followed by the probability

of our preferred outcome

in parenthesis.

Therefore,

we read the following statement as

variable 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 only have one 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 U.S. elections.

Usually,

when dealing with a Bernoulli distribution,

we either have the probabilities

of one of the events occurring,

or have past data indicating

some experimental probability.

In either case,

the graph of a Bernoulli distribution is simple.

It consists of two 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 one minus P.

For Bernoulli distributions,

we often have to assign which outcome is zero

and which outcome is one.

After doing so,

we can calculate the expected value.

Bear in mind that depending

on how we assign the zero and the one,

our expected value will be equal

to either P or one minus P.

We usually denote the higher probability

with P and the lower one with one minus P.

Furthermore, conventionally,

we also assign a value

of one to the event with

the probability equal to P.

That way the expected value

expresses the likelihood

of the favored event.

Since we only have one trial

and a favored event,

we expect that outcome to occur.

By plugging in

P and one minus P

into the variance formula,

we find that the variance

of Bernoulli events would always

equal P times one 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 called unfair

because its weight is spread disproportionately,

and it gets tails 60% of the time.

We assign the outcome

of tails to be one 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.

Good job, everybody.

Sometimes,

instead of wanting to know which

of two outcomes is more probable,

we want to know how often it would

occur over several trials.

In such cases,

the outcomes follow a binomial distribution,

and we will explore it further in the next lecture.