Mean of sum and difference of random variables

- [Instructor] Let's say that
I have a random variable X

which is equal to the number
of dogs that I see in a day.

And random variable Y is
equal to the number of cats

that I see in a day.

Let's say I also know what
the mean of each of these

random variables are, the expected value.

So the expected value of X
which I could also denote

as the mean of our random
variable X let's say I expect

to see three dogs a day
and similarly for the cats,

the expected value of Y is
equal to I could also denote

that as the mean of Y is going
to be equal to and this is

just for the sake of (mumbles)
let's say I expect to see

four cats a day.

And pretty much we define how
you take the mean of a random

variable or the expected
value for a random variable.

What we're going to think
about now is what would be

the expected value of X plus
Y or other way of saying

that the mean of the sum of
these two random variables.

Well it turns out, and I'm
not proving it just yet,

that the mean of the sum of
random variables is equal to

the sum of the means.

So this is going to be equal
to the mean of random variable

X plus the mean of random variable Y.

And so in this particular case,
if I were to say well what's

the expected number of dogs
and cats that I would see

in a given day.

Well I would add these two
means, it would be three plus

four it would be equal to seven,
so in this particular case

it is equal to three plus
four which is equal to seven.

And similarly if I were to ask
you the difference if I were

to say how many more cats in
a given day would I expect

to see than dogs, so the
expected value of Y minus X.

What would that be?

Well intuitively you might
say well hey if we can add

random...

If the expected value of the
sum is the sum of the expected

values, then the expected
value or the mean of

the difference will be the
differences of the means

and that is absolutely true.

So this is the same thing as
the mean of Y minus X which is

equal to the mean of Y is going
to be equal to the mean of Y

minus the mean of X, minus the mean of X.

And in this particular case,
it would be equal to four

minus three, minus three is equal to one.

So another way of thinking about
this intuitively is I would

expect to see on a given
day one more cat than dog.

The example that I just
used this is discrete random

variables, on a given day I
wouldn't see 2.2 dogs or pi

dogs, the expected value itself
does not have to be a whole

number 'cause you could of
course average it over many days.

But this same idea that the
mean of a sum is the same thing

as a sum of means and the
mean of a difference of random

variables is the same as
the difference of the means.

In a future video I'll do a proof of this.