When we are investigating probability density functions, it is very useful to be able to describe them
using a few key properties and these properties can be the main, the variance and the skew.
Now, we've already looked at using the expectation of radar to calculate the main of the distribution,
but it can also be used to calculate these other useful properties as well.
He we have a probability density function, and you can see from the distribution that most of the likelihood
area is in this area here.
So if we were to calculate the mean of the distribution will end up with a man looking like this.
And in the past, we've shown that we can use the expectation of radar here to calculate the mean of
the distribution.
Another useful way to describe a distribution is by its variance, so this distribution here can be
said to have quite a high variance compared to the second distribution here.
So this distribution here is a low variance because it's more squashed in.
The variance is a measure of how far away the distribution spread away from the main.
The variance of the distribution can be calculated using this equation up here.
So this equation here basically takes the difference between the random variable and its main squares
it and works out the main value of that whole term.
So it's a measure of the spread of the distribution.
Now, this equation can also be rewritten b b range by expanding it out and written in this form here.
So you can basically say we're working out the expected value of the random variable squared minus the
main square.
It's going to give you the variance.
So the variance is equal to the standard deviation squared.
So when we have a square here, we can we can say it's a variance if it's just the single sigma here.
This is just a standard deviation.
So using the mean and the variance, we can describe a distribution and it's fairly common to write
a distribution such as like this.
So here this just means that we have X, which is a random variable.
The distribution has a mean of exposure and a variance of this sigma squared here.
The last property that we're going to look at is the skewness of the distribution.
So this is also another useful way of describing the shape of the distribution.
So, again, looking at this distribution here, this is a fairly symmetric distribution.
So we have the main value here.
And you can see that on both sides of it.
It's pretty much the same shape, just married.
So it is a very symmetrical distribution.
If we can compare this distribution to this distribution chain here, you can see that it's no longer
symmetrical.
So now it's a lot more asymmetrical on one side than the other.
So the Skewness is a measure of how asymmetrical the distribution is.
More specifically, the skewness of a distribution is a measure of the asymmetry of the distribution
from the main.
And again, it can be calculated using this equation here.
So instead of the variance, we taking the difference from the main and we're doing it to the cube instead
of the square.
So this is going to give us the skewness of the distribution.
Now, this year, the skew is also very difficult to compare between distributions.
So what we normally do is that we normally calculate the coefficient of Skewness, which is just equal
to the ski value divided by the standard deviation of the distribution to the power of three.
So to recap, the three different properties that are usually used to describe the distribution are
the main, which can be be considered as the first moment.
The variance, which can be considered as the second moment, as you can see here, is to the power
of two and the SKU, which is usually considered the third moment because we have the DePaolo three.
Now, there's also higher moments, but they become a bit more abstract and not as useful at describing
a probability distribution.