In the last video we learned a little bit about what the
expected value of random variable is, and we saw that it
was really just the population mean-- the same thing.
But with a random variable, since the population is
infinite, you can't take up all of the terms and
then average them out.
What you have to do is say OK, each of these terms occur with
some frequency or with some probability, and you kind of
just take a probability weighted sum.
Which we saw in the last video was the exact same thing as
adding everything together and dividing by the number of
numbers, except that that method worked with an infinite
number of an infinite population what the
random variable is.
Because you can just keep on performing the experiment that
generates the random variable.
And then, we actually calculated the expected value
for the particular binomial distributions that we studied,
especially the one with the flipping of the coin.
In this video we'll find a general formula for the mean,
or actually, for the expected value of a binomial
distribution.
So if we say that the random variable, x, is equal to the
number of-- we could call it successes.
The number of successes with probability p after n trials.
So I'm being a little bit generally here.
I mean I could have said the number of successful heads,
which have a probability of 0.5 after 10 flips.
That's the same thing is this, I'm just being a little
bit more general here.
And now, we'll actually figure out what the
expected value of this is.
And we saw if you actually figured out the probability
distribution for this random variable you get that nice
binomial distribution that looks a little bit
like a bell curve.
And we'll study more about bell curves later.
And before I actually show it to you I'll
give you the answer.
Because the answer, to some degree, is actually
pretty intuitive.
The expected value of this random variable is n times
p, or sometimes people will write p times n.
Let me make that a little bit more tangible for you.
So if I said that x is-- let me do it in a different color.
Let's see, x is equal to the number of baskets I make.
Where I'm talking about basketball, not basket weaving.
Number of baskets I make after 10 shots where I have the
probability of me making any 1 shot is-- I don't know-- 40%.
We know that the expected number of baskets I
make after 10 shots.
So we know that the expected number of baskets I make after
10 shots, where each of my shots is 40%-- all I have to do
is multiply the probability times the number of
baskets I'm taking.
So I multiply probability times the number of baskets or the
number of shots I'm taking, which should be equal to 4.
So I know I said-- and you really shouldn't necessarily
strictly view expected value as the number of shots you should
expect to make because sometimes probability
distributions can be kind of weird.
But in the binomial distribution you can kind
of view it that way.
That this is the number of shots you would expect to make.
Or you can kind of view it as the most likely outcome.
That if you have a 40% shot percentage, and you takes 10
shots, the most likely outcome is that you'll make 4 shots.
You still might make 6 shots or 3 shots, but this is going to
be the most likely outcome.
and in my head, the way I think about, the way it makes
intuitive sense is that every time you shoot you have a 40%
chance of making the shot.
So you could say that you always make 40% of a shot.
And if you take 10 shots you're going to make 4 whole shots.
So that's one way to think about it and why this might
make a little intuitive sense.
But now, let's prove it to ourselves that this is really
true for any a random variable that's described by a
binomial distribution.
So in a binomial distribution what is the probability-- so if
I say, what is the probability that X is equal to k?
And I know it just gets a little complicated sometimes.
But I'm just saying, what's the probability say in
this basketball analogy.
Would be you know, what's the probability that I make--
k could be 3 shots or something like that.
So that's what we're talking about.
And that we learned was, if we're taking n shots we're
going to choose k of them.
And we did that several times in the last couple of videos.
And then we multiply that times the probability of any one of
those particular occurrences.
So if I'm making k shots, it'll be the probability of me making
any one shot, Which is p to the kth power.
p times itself k times.
That's the probability of me making k shots.
And then the rest of the shots I have to miss.
So the probability of a miss is 1 minus p.
And then how many shots?
If I've made k shots, the rest of the shots I have to miss.
So I'm going to miss n minus k shots.
So in any binomial distribution this is a probability
that you get k successes.
Now we know that the expected value, the way you calculate an
expected value of a random variable is you just take the
probability weighted sum.
I don't want to confuse you too much and if you main take
away from this video is just this, that's good enough.
You should feel good.
Now it'll get a little technical, but it'll hopefully
make you a little bit more comfortable with sigma and
sum notation as well.
It'll make you a little bit more comfortable with
binomial coefficients and things like that.
But just going back, the expected value is a a
probability weighted sum of each of these.
So what you want to do is you want to take the probability
that X is equal to k, times k, and then add that up for
each of the possible k's.
So how would I write that?
So the expected value of X, the expected value of our random
variable that's being described as binomial distribution--
it's equal to the sum.
And we're going to sum all of the values that k can take on.
So k can start at 0-- in the basketball version, I make no
shots-- all the way to n, which means I make n shots.
And for each of them you want to multiply k, so the outcome,
so I made k shots, times the probability that
I make k shots.
Well, what was the probability that I make k shots?
That was this right here.
So it'd be k times n choose k times p to the k times 1
minus p to the n minus k.
And now we're just going to do some algebra, some
sigma algebra I guess you could call it.
So the first simplification we can make is we're summing
from k equals 0 to n.
So the first term here is going to have a k equals 0 here.
This is going to be 0 in the first term.
So this first term is 0, then this whole thing is going to be
0, and the k equals 0 term won't contribute to the sum
because this whole thing will be 0.
Let me that write because I think it's-- so this sum could
be written as 0 times n choose 0 times p to the 0 times 1
minus p to the n minus 0.
Plus 1 times n choose 1 times p to the 1 times 1 minus
p to the n minus 1.
And then you keep adding, all the way until you
get to k is equal to n.
So it'd be n times n choose n times p to the n, times
1 minus p, n minus n.
This is just another way of writing this sum up here.
And what I just said is the first term, which is this term,
is going to be equal to 0 because k is equal to 0.
0 times any thing is 0.
So we can ignore that term and we can rewrite this sum as
essentially this sum right here.
And so if we were to do that we're essentially just
rewrite this thing up here.
So the expected value of our random variable
is equal to the sum.
And we don't have to go from k equal 0, we could
start at k equal 1 now.
From k equals 1 to n of the same thing. k times n choose
k times p to the k, times 1 minus p, n minus k.
Let's see what we can do from here.
All we did so far is we got rid of that first term because
that's kind of the trick we'll use to simplify this eventually
to the result that we want to have.
So let's write out the binomial coefficient and see if we
can do something there.
Oh, look at this.
My iPod wants to sync.
Let me get rid of that.
All right, so then, where was I?
OK, so then this is equal to-- I'm just going to write out
the binomial coefficient.
k equals 1 to n.
k times-- this right here is n factorial over k factorial
over n minus k factorial.
Times p to the k times 1 minus p to the n minus k.
And here we can make a little bit of a simplification
because what's k divided by k factorial?
Maybe I could rewrite it a different way. k factorial is k
times k minus 1 times k minus 2, and so forth, all the
way until you get to 1.
This is k factorial.
So k factorial could be written as k times k minus 1 factorial.
It's k times, and then the number 1 smaller then k times
all the numbers below it.
So let me rewrite.
So this could be rewritten as k times k minus 1 factorial.
And the whole reason why I did that is so I can cancel
this k out with that k.
So if I cancel that out I think this warrants rewriting
the whole thing again.
So now, I guess you could argue simplified it to, it equals the
sum from k is equal to 1 to n of n factorial over k
minus 1 factorial.
Times n minus k factorial times p to the k times 1 minus
p to the n minus k.
And let's do another simplification.
Now, what I want to do and we kind of know where
we're heading, right?
This should simplify to n times p.
So let's see if we can factor out an n times p and then let's
see if we can turn everything else into a 1, and then
we would be done.
So we could rewrite n factorial using the same trick up here.
n factorial can be rewritten as n times n minus 1 factorial
by the same logic.
And then p to they k could be rewritten as p times
p to the k minus 1.
And then we can factor out this n and this p and we'll get it's
equal to np times the sum from k is equal to 1 to
n of-- let's see.
We factored that n and p out.
n minus 1 factorial over k minus one factorial times
n minus k factorial.
Times p to the k minus 1.
That's not in the denominator.
This is just a regular-- times 1 minus p to the n minus k.
And we're close.
Remember, we want the result that the expected value of our
variable, and that's what we were doing before.
That this should be equal to this.
So we'll be done if we can just show that this whole
thing here equals 1.
And to do that I'll make a simplifying substitution.
Let me make the substitution-- I don't know-- let's say that
a is equal to k minus 1.
And that b is equal to n minus 1.
And then what would n minus k equal to?
Let's see.
If a is equal to k minus 1 then a plus 1 is equal to k.
And then here, b plus 1 is equal to n, so then n minus
k would be equal to this, a plus 1 minus this.
Minus b minus 1, these would cancel out.
Which would equal a minus b.
And let's see if we can simplify this.
So this whole sum will then turn into np times the sum
from-- OK, when k is equal to 1, that's the same thing--
when k is equal to 1, what is a equal to?
a is equal to 0.
From a equal to 0 to-- now when k is equal to n,
what will a be equal to?
If this is equal to n, if k is n, then a is
equal to n minus 1.
So we have a equal to a to a equals n minus 1.
But n minus 1 is the same thing as b.
So we could rewrite the sum there.
That's always a little confusing.
You might want to pause and think about that a little bit.
But I realize I'm already over time, so I'll just
keep moving forward.
And then we had b is equal to n minus 1.
So that'll be b factorial over k minus 1 , we made
the definition that that's equal to a.
So that's a factorial.
And then over here, n minus k should be a--
oh, you know what?
I reversed this. n minus ok should be b minus a.
n minus k-- right.
n is b plus 1, so it's b plus 1 minus a plus 1.
Minus a, minus 1.
So the 1's cancel out and you get b minus a.
So the n minus k will become b minus a factorial.
And then p to the k minus 1-- k minus 1 is p to the a.
And then times 1 minus p to the n minus k.
We already showed that n minus k is the same
things as b minus a.
And then here, and we're pretty much done now-- this
right here, what is this?
This is the probability of-- well, let me rewrite
it a simpler way.
This is equal to np times the sum from a is equal to 0 to b.
What is this?
This is b choose a.
I have b things and I want to choose a things from them, how
many different ways can I-- times p to the a times 1
minus p to the b minus a.
What is this thing here?
This is you're taking every term of the binomial
distribution.
So you're saying, what is the probability
when a is equal to 0?
This is the probability for each of the a's, right?
And you're summing up over all of the a's you can achieve.
So if I were to draw a quick dirty distribution like this,
if a is equal to 0 you have a certain probability.
Then a certain probability for a equals 1. and then another
probability, and it goes higher.
Then it's like a bell curve, something like that.
This term right here is each of these.
Each of these boxes you could say represents
one of these terms.
When a equals 0 it's this term.
When a equals 1 it's this term.
When a equals 2 it's this term, all the way to b terms.
But we're summing them up, so we're summing all
of the probabilities.
We're summing over all of the values that our
random variable can take.
So if we solved all the probabilities that a random
variable can take, or we're summing over all of the values,
this is going to sum up to 1.
This is like saying that this is the probability of heads
plus the probability of tails.
Or you could say in the flipping of a coin analogy,
this is the probability that I have one heads plus the
probability I have 2 heads, plus the probability I have 3
heads, plus the probability I have 4 heads, all the way to
the probability I have b heads.
So it's pretty much every circumstance that can occur.
So this is the sum over the entire probability
distribution, and so that is going to be equal to 1.
And so, we're left with the expected value of our
random variable, X, is equal to n times p.
Where n is the number of trials and p is the probability
of success on each trial.
And this is true only for binomial distributions.
It isn't true for any random variable, X.
Only true for random variable, X, whose probability
distribution is the binomial distribution.
Anyway, I'm all out of time.
I'll see you in the next video.