-
-
In the last video we figured
out the mean, variance and
-
standard deviation for our
Bernoulli Distribution with
-
specific numbers.
-
What I want to do in this video
is to generalize it.
-
To figure out really the
formulas for the mean and the
-
variance of a Bernoulli
Distribution if we don't have
-
the actual numbers.
-
If we just know that the
probability of success is p
-
and the probability a failure
is 1 minus p.
-
So let's look at this, let's
look at a population where the
-
probability of success-- we'll
define success as 1-- as
-
having a probability of p, and
the probability of failure,
-
the probability of failure
is 1 minus p.
-
Whatever this might be.
-
And obviously, if you add these
two up, if you view them
-
as percentages, these are
going to add up to 100%.
-
Or if you add up these
two values, they are
-
going to add to 1.
-
And that needs to be the case
because these are the only two
-
possibilities that can occur.
-
If this is 60% chance of success
there has to be a 40%
-
chance of failure.
-
70% chance of success, 30%
chance of failure.
-
Now with this definition of
this-- and this is the most
-
general definition of a
Bernoulli Distribution.
-
-
It's really exactly what we did
in the last video, I now
-
want to calculate the expected
value, which is the same thing
-
as the mean of this
distribution, and I also want
-
to calculate the variance, which
is the same thing as the
-
expected squared distance of
a value from the mean.
-
So let's do that.
-
So what is the mean over here?
-
What is going to be the mean?
-
Well that's just the probability
weighted sum of
-
the values that this
could take on.
-
So there is a 1 minus p
probability that we get
-
failure, that we get 0.
-
So there's 1 minus
p probability of
-
getting 0, so times 0.
-
And then there is a p
probability of getting 1,
-
plus p times 1.
-
Well this is pretty
easy to calculate.
-
0 times anything is 0.
-
So that cancels out.
-
And then p times 1 is
just going to be p.
-
-
So pretty straightforward.
-
The mean, the expected value
of this distribution, is p.
-
And p might be here
or something.
-
So once again it's a value that
you cannot actually take
-
on in this distribution,
which is interesting.
-
But it is the expected value.
-
Now what is going to
be the variance?
-
What is the variance of
this distribution?
-
Remember, that is the weighted
sum of the squared distances
-
from the mean.
-
Now what's the probability
that we get a 0?
-
We already figured that out.
-
There's a 1 minus p probability
that we get a 0.
-
So that is the probability
part.
-
And what is the squared distance
from 0 to our mean?
-
Well the squared distance from
0 to our mean-- let me write
-
it over here-- it's going to be
0, that's the value we're
-
taking on-- let me do that in
blue since I already wrote the
-
0-- 0 minus our mean-- let
me do this in a new
-
color-- minus our mean.
-
That's too similar
to that orange.
-
Let me do the mean in white.
-
0 minus our mean, which is p
plus the probability that we
-
get a 1, which is just p-- this
is the squared distance,
-
let me be very careful.
-
It's the probability weighted
sum of the squared distances
-
from the mean.
-
Now what's the distance-- now
we've got a 1-- and what's the
-
difference between
1 and the mean?
-
It's 1 minus our mean, which
is going to be p over here.
-
And we're going to want to
square this as well.
-
This right here is going
to be the variance.
-
Now let's actually
work this out.
-
So this is going to be
equal to 1 minus p.
-
Now 0 minus p is going
to be negative p.
-
If you square it you're just
going to get p squared.
-
So it's going to be p squared.
-
Then plus p times-- what's
1 minus p squared?
-
1 minus p squared is going to be
1 squared, which is just 1,
-
minus 2 times the
product of this.
-
So this is going to be minus
2p right over here.
-
And then plus negative
p squared.
-
So plus p squared
just like that.
-
And now let's multiply
everything out.
-
This is going to be, this term
right over here is going to be
-
p squared minus p
to the third.
-
And then this term over here,
this whole thing over here, is
-
going to be plus
p times 1 is p.
-
p times negative 2p is
negative 2p squared.
-
And then p times p squared
is p to the third.
-
Now we can simplify these.
-
p to the third cancels out
with p to the third.
-
And then we have p squared
minus 2p squared.
-
So this right here becomes,
you have this p right over
-
here, so this is equal to p.
-
And then when you add p squared
to negative 2p squared
-
you're left with negative p
squared minus p squared.
-
And if you want to factor a p
out of this, this is going to
-
be equal to p times, if you take
p divided p you get a 1,
-
p square divided by p is p.
-
So p times 1 minus p, which is
a pretty neat, clean formula.
-
So our variance is p
times 1 minus p.
-
And if we want to take it to the
next level and figure out
-
the standard deviation, the
standard deviation is just the
-
square root of the variance,
which is equal to the square
-
root of p times 1 minus p.
-
And we could even verify that
this actually works for the
-
example that we did up here.
-
Our mean is p, the probability
of success.
-
We see that indeed it
was, it was 0.6.
-
And we know that our variance is
essentially the probability
-
of success times the probability
of failure.
-
That's our variance
right over there.
-
The probability of success
in this example was 0.6,
-
probability of failure
was 0.4.
-
You multiply the two, you get
0.24, which is exactly what we
-
got in the last example.
-
And if you take its square
root for the standard
-
deviation, which is what we
do right here, it's 0.49.
-
So hopefully you found that
helpful, and we're going to
-
build on this later on in some
of our inferential statistics.
-