The expectation of radar allows us to calculate what the main or average or the expected value is or
a random variable, if it is tested over a large number of experiments, we can see why this would be
useful later on.
But for now, let's suppose that we have an experiment with a discrete random variable X and we're going
to test this variable in times.
Every time we do this, we observe the value of a one and one times the value of a two and two times
and so forth, all the way up to the value of AM occurring and M times the expected value of this random
variable X can be computed using this equation shown below.
So this equation basically sums up the expected value times, the number of times that happens over
the whole sample set and divided by the number of tests to get the expected value.
So now let's use a Disick sample again.
And suppose we roll the dice and times and we count and we count the number of times each number is
shown.
We should end up with a table that looks like this.
So here on this side here, we have the different DI numbers from one to six and from all the different
tests, the 20 tests, this is how many times the one is shown.
This is how many times of the two is shown and so forth.
Now we can use this type of data and the expectation equation up here to calculate what the expected
value is.
So we substitute the numbers into equation with the equation that looks like this segment C, we have
one error.
And so one of the 20 we have the value DI one and we occur twice the value of two occurs four times
and so forth, all the way up to the value of six, which occurs four times as well.
So if we do this equation and we calculate that the expected value is three point eighty five.
Now you may notice the value three point eighty five is not actually any of the expected outcomes in
this great set.
But the expected value does not actually have to be in in the set.
It's just the overall average number from all the experiments.
So let's take this example to the extreme.
Let's assume that we spend the rest of our lives rolling the dice.
What is the actual expected value of the DI then?
Well, we know that the probability of rolling in any number is one of the six.
And from a basic probability, we know so that if we roll the dice and times we expect to see and divided
by six of each number.
So in over six ones and of a six or two's and so forth.
So we can use this and we can write it into the expected value equation.
So we come up with an equation like this.
So for if we run the experiment in times, we should expect in over six ones and over six of twos and
so forth.
And then if we run this experiment and let intends towards infinity, so if we just keep rolling dice
over and over and over again, we can actually calculate the the expected value is three point five.
So we can see that the experiment results are fairly close to what the actual mathematical results are.
So for discrete random variable, we can use the equation that we've shown before.
So now let's extend this concept to continuous random variables and the probability density function.
We can use a probability density function if x to calculate the expected value using this equation here.
So it's very similar.
It's over a summation.
We're using the interval between the two absolute limits of negative energy, infinity.
And we have the value of the random variable X, which is this replaces the eye here and we actually
have the number of times it occurs and it's just going to be the probability.
So you can see that the integral of F of X, the X, that's the area on the curve.
So that's going to be the probability.
So this equation here is simply calculating the main value that we can expect of the random variable
if we test every single possibility, infinite times, the expected value or main value of the random
variable X can be written simplistically using a double annotation.
So we have the random verbal X bar or we could have the little X bar.
So this is different ways of writing that we want to use the mean or the expected value of the random
variable.
Suppose that we ran an experiment on a random variable with a given probability density function multiple
times and a certain random number shows up more times than any other random number from the set.
It just must mean that this number has a higher probability in a density function and the other the
reverse is also true.
If a certain number has a higher probability than the rest, then it's more likely to show up for any
given experiment.
So we can use the expectation of radar to try to quantify this.