L06.3 The Variance of the Bernoulli & The Uniform

L06.3 The Variance of the Bernoulli & The Uniform


In this segment, we will go
through the calculation of the variances of some familiar
random variables, starting with the simplest one that we
know, which is the Bernoulli random variable. So let X take values 0 or 1, and
it takes a value of 1 with probability p. We have already calculated the
expected value of X, and we know that it is equal to p. Let us now compute
its variance. One way of proceeding is to use
the definition and then the expected value rule. So if we now apply the expected
value rule, we need the summation over all possible
values of X. There are two values– x equal to 1 or x equal to 0. The contribution when X is
equal to 1 is 1 minus the expected value, which
is p squared. And the value of 1 is taken
with probability p. There is another contribution
to this sum when little x is equal to 0. And that contribution is going
to be 0 minus p, all of this squared, times the probability
of 0, which is 1 minus p. And now we carry out
some algebra. We expand the square here, 1
minus 2p plus p squared. And after we multiply with this
factor of p, we obtain p minus 2p squared plus p
to the third power. And then from here we have a
factor of p squared times 1, p squared times minus p. That gives us a minus p cubed. Then we notice that this term
cancels out with that term. p squared minus 2p
squared leaves us with p minus p squared. And we factor this as
p times 1 minus p. An alternative calculation
uses the formula that we provided a little earlier. Let’s see how this will go. We have the following
observation. The random variable X squared
and the random variable X– they are one and the same. When X is 0, X squared
is also 0. When X is 1, X squared
is also 1. So as random variables, these
two random variables are equal in the case where X
is a Bernoulli. So what we have here is just the
expected value of X minus the expected value of X squared,
to the second power. And this is p minus p squared,
which is the same answer as we got before– p times 1 minus p. And we see that the calculations
and the algebra involved using this formula
were a little simpler than they were before. Now the form of the variance
of the Bernoulli random variable has an interesting
dependence on p. It’s instructive to plot
it as a function of p. So this is a plot of the
variance of the Bernoulli as a function of p, as p ranges
between 0 and 1. p times 1 minus p
is a parabola. And it’s a parabola that is
0 when p is either 0 or 1. And it has this particular
shape, and the peak of this parabola occurs when p is equal
to 1/2, in which case the variance is 1/4. In some sense, the variance is
a measure of the amount of uncertainty in a random
variable, a measure of the amount of randomness. A coin is most random if it
is fair, that is, when p is equal to 1/2. And in this case, the variance
confirms this intuition. The variance of a coin flip is
biggest if that coin is fair. On the other hand, in
the extreme cases where p equals 0– so the coin always results in
tails, or if p equals to 1 so that the coin always results in
heads– in those cases, we do not have any randomness. And the variance, correspondingly, is equal to 0. Let us now calculate
the variance of a uniform random variable. Let us start with a simple case
where the range of the uniform random variable starts
at 0 and extends up to some n. So there is a total of n plus 1
possible values, each one of them having the same
probability– 1 over n plus 1. We calculate the variance using
the alternative formula. And let us start with
the first term. What is it? We use the expected value rule,
and we argue that with probability 1 over n plus 1, the
random variable X squared takes the value 0 squared, with
the same probability, takes the value 1 squared. With the same probability, it
takes the value 2 squared, and so on, all of the way
up to n squared. And then there’s
the next term. The expected value of the
uniform is the midpoint of the distribution by symmetry. So it’s n over 2, and we take
the square of that. Now to make progress here, we
need to evaluate this sum. Fortunately, this has
been done by others. And it turns out to be equal
to 1 over 6 n, n plus 1 times 2n plus 1. This formula can be proved by
induction, but we will just take it for granted. Using this formula, and after a
little bit of simple algebra and after we simplify, we obtain
a final answer, which is of the form 1 over
12 n times n plus 2. How about the variance
of a more general uniform random variable? So suppose we have a uniform
random variable whose range is from a to b. How is this PMF related to the
one that we already studied? First, let us assume that n
is chosen so that it is equal to b minus a. So in that case, the difference
between the last and the first value of the
random variable is the same as the difference between the last
and the first possible value in this PMF. So both PMFs have the same
number of terms. They have exactly
the same shape. The only difference is that the
second PMF is shifted away from 0, and it starts at a
instead of starting at 0. Now what does shifting
a PMF correspond to? It essentially amounts to taking
a random variable– let’s say, with this PMF– and adding a constant to
that random variable. So if the original random
variable takes the value of 0, the new random variable
takes the value of a. If the original takes the value
of 1, this new random variable takes the value
of a plus 1, and so on. So this shifted PMF is the PMF
associated to a random variable equal to the
original random variable plus a constant. But we know that adding
a constant does not change the variance. Therefore, the variance of this
PMF is going to be the same as the variance of the
original PMF, as long as we make the correspondence that
n is equal to b minus a. So doing this substitution in
the formula that we derived earlier, we obtain 1 over
12 b minus a times b minus a plus 2.

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