In this video, we're going to look at probability density functions now, mathematics is a very precise
and deterministic field, so a way of expressing random events and uncertainties is required.
A random variable is a way to mathematically express steady acoustic outcomes as real numbers so we
can write a random variable like say so X is the random variable and it has a possibility of being any
of the real numbers in set a given all the different outcomes of an experiment in this set is so random,
variable is simply a mapping of all the experimental outcomes or events into real numbers of that e..
However, once the experiment has been carried out, the value of the random variable or outcome has
been determined.
Once this happens, you can then treat the value of the random variable as just a normal algebraic value.
Let's say we have a random variable X, but we like to know how likely each outcome is to occur, is
each outcome equally likely to occur?
So if we're talking about a crossover, TOYIN is 50/50 for either sides or some outcomes more likely
to occur than others?
We describe and quantify this with probability density functions.
So probability density functions of PDF measure the relative likelihood or probability that a specific
outcome will occur.
So a probability density function might look like, say, on the y axis, we have the likelihood and
on the x axis over here, we have the different values that the random variable can take.
So in this case, the random variable can take any number of very negative Vendy and Infinity.
It's just a real plane this way.
And this red line here would actually be the probability density function.
And we commonly write this as little ethics as a function of X.
So the value of this function at each point is the likelihood of that event occurring.
However, it is important to note that likelihood and probability are not the same thing, we can calculate
a probability from the likelihood function, but only when we consider it from between a range of values.
So to calculate the probability that the random variable would take a number between two values, let's
say between A and B. We can use this equation here.
So the probability that the random variable X is going to be between A and B is just the integral between
A and B of the probability density function.
So this is basically saying the area under the curve is a probability.
So if we were to integrate the whole curve between negative infinity and infinity, by definition,
the probability has to equal to one.
So if we want the probability that the random variable value between zero and infinity, we have to
calculate this area here, the area on the curve for the positive x axis.
So let's have a look at a more concrete example, let's say we have a probability density function that
looks like this, so we have an even probability that we'll get a random variable number between negative
one and one.
So we're basically saying the likelihood value is just a constant value.
So if X is just going to equal 2.5 between negative one and one and everywhere else, it's going to
be zero.
So let's say we want to know the probability that the random variable X is going to take on a value
of negative point seventy five.
The answer is actually just going to be zero because we want to look at the area under the curve.
So if we just try to evaluate the likelihood function at a single point such as this point here, it
has zero area because they are only looking at a single point to calculate the probability we actually
have to take the integral.
So now let's say we want to know the probability that the random variable will be between negative one
and negative point seventy five.
Well, then we can just calculate the area of the square here so we know the value is point five here.
We know the distance is O point to five.
So therefore the area is just zero point one two five.
So the probability that the random variable X takes a value between negative one and negative point
seventy five is just this area here.
So this is three point twenty five multiplied by point five.
So we get three point one, two five.
So this is how we can use the probability density functions to work out probability, so the function
graphs likelihood and the area under the graph is probability.
Now, the probability of the whole area has to equal one, because we know that out of all the cases,
the random variable has to take on a single value.
Now that we have defined what a payday for a random variable is and how we can calculate the probability
of certain events happening, a cooperation that can be applied to any PDF is the expectation operator
and that's given by this equation here.
So basically the expectation operator.
So E of X is just the integral of the whole density function.
So from negative Phinney to infinity of the random variable X times the probability density function.
To give the expectation operator a bit more meaning, this is basically calculating the main of the
probability density functions.
So the expected value or main of the random variable is usually for simplicity, just as this X bar
or little expo is just the expected value of the random variable probability density function.
So the expectation over it allows us to investigate some distribution statistical properties.
It is very useful to be able to describe a probability density function or distribution in a few key
properties.
And the first one we've already covered, which is the main and some other common parameters, are the
variance and the skew.
And each of these distribution statistics can be calculated using the expectation operator.
So first, let's have a look at the main, so let's say we have a probability density function that
looks like this.
Now, the main is just going to be the average of the distribution.
So in this case, it's going to be at this point here and we can say that this is X, so this is the
main or the random variable distribution and simply just calculated from the expectation operator of
the random variable.
The distribution variance looks at how spread the distribution is, so you can see on this distribution
here, we can say it's got quite a large variance, is quite large compared to a distribution that has
a smaller variance, is more tightly compact.
So we call this parameter the variance and it's basically a measure of the spread of the distribution.
So this is an Sigma X, so the variance of a distribution is a measure of how much the distribution
varies away from the main value here and again is calculated using the expectation of radar.
So using the expectation operator, we calculate the random variable distribution away from the main
and we square it.
And this is going to give us the variance or the sigma squared, the standard deviation squared.
We can also expand this top equation and write the variance in a slightly different form we get the
variance is just going to be the expected value squared minus the main squared.
The main and the variance give us quite a bit of information about the distribution and effect is going
to be the two main parameters when we're looking at the Gaussian or normal distribution.
So we could actually write the distribution to be random variable.
X is a distribution that has a main export and variance sigma squared X.
And then lastly, we can look at the distribution skewness, so the skewness of the distribution looks
about how symmetrical is.
So in this case, we have a very symmetrical distribution compared to this case where we have a non-special
distribution, we have a very long tail on one side.
So this distribution is going to be more skewed towards one way than the other.
And we can calculate the Skewness by basically looking at the expectation of radar for the random variable
difference away from the main but raised to the power of three instead of the power to for the variance
so we can calculate the skew as the expectation of rate.
But we can also look at the coefficient of skew or the skewness, just the value of skew divided by
the standard deviation to the power of three.
So the three main statistical properties of a distribution can be the main, which is just the expectation
of radar, which is also commonly called the first moment.
The variance or second moment is just a difference away from the main square.
And then the skew is the third element, which is just the difference away from the main to the power
of three.
In this video, we have quickly covid what random variables are, how we can describe the distribution
or random variables, so how we can describe how likely certain values of the variable are.
And then we looked at statistical properties to describe these distributions.