So, those are our calculations for the key summaries, and

just to do a particular example, what if we have p = 0.5?

So that's a very special instance of the Bernoulli,

where each outcome is equally likely.

That's tossing a fair coin.

Then you'll find that for p = 0.5,

the mean is just equal to 0.5 itself, remember the mean is equal to p.

The variance is p times 1-p, a half times a half is a quarter,

and then the standard deviation if you square root the quarter,

you are going to get a half, and so, those are the key summaries.

And, this process of creating the summaries is very useful as I say

at the end of having done something like a Monte Carlo simulation to say,

here's why I think the answer is going to be mu and then we can use the standard

deviation to give us a sense of the variability about the mean.

So here's a example of where this Bernoulli distribution could come

in useful.

And I'm going back to one of the initial examples of this class,

where we're talking about drug development and creating a probabilistic model

would potentially involve these Bernoulli random variables.

So we might come up with a scenario where we think the outcome of whether or

not a drug is approved by some regulatory body is a Bernoulli random variable.

It's either approved yes or it's not approved, no.

We can write those as the values 1 and 0 so we're back to this Bernoulli setup.

And then our experts might have estimated that the probability that we're a yes,

that we get approved, is 0.65 and the probability we don't get approved is 0.35.

Notice those two add up to one.

Now if we believe that given the drug is approved,

the projected revenue is $500 million from it, zero otherwise because

if it's not approved we can't sell it, so we'll get no revenue at all.

We could work out the expected revenue.

And the calculation of these expectations is as before you take the different

realizations of the random variable and the revenues under each realization.

More applied through the probabilities of those events.

So if we get approved,

which is probability 0.65, we get 500 million revenue.

And if we don't get approved we get zero revenue, and

that happens with probability 0.35.

We add those up and we get an expected revenue of $325 million.

And so this Bernoulli could potentially be used as a building block for

a Monte Carlo simulation to get a sense of,

if we've got ten of these drugs what's the expected revenue going to be?