So it's the area from minus infinity to x of our probability density function.

It'll actually be the integral from 4 and a half to 5 and a half of this probability density function or of this probably density function, the x.

You know what the standard deviation means in general but this is the standard deviation of this distribution, which is a probability density function.

K 0 calculates the density function K 1 calculates the distribution.

Cumulative 0 calculates the density function cumulative 1 calculates the distribution.

So a good chunk of unsupervised learning seeks to recover the underlying the density of probability distribution that generated the data.

So you just learned about the concept of a probability density and that's very cool.

But in a continuous probability distribution or a continuous probability density function, you can't just say what is the probability of me getting a 5.

And, of course, what is the volume of that column?

For those of you who know calculus, if p of x is our probability density function doesn't have to be a normal distribution although it often is a normal distribution the way you actually figure out the probability, let's say between 4 and a half and 5 and half.

We know that the density for the birth time to be before noon is twice as large as the density for being born after noon.

While an orbital is actually a mathematical probability function that tells us where we're

And it's kind of like a probability for continuous spaces but not exactly.

Or decision making under uncertainty.

Models of primardial nucleosynthesis have been developed with grea t precision and now they're compared with observations.

So, given what we learnt, you want to assign 0 to anything outside that range but inside this range, we'd love to give a value and say,

So let's say the density in this volume that we care about, the density function is a function of x, y and z.

Let's think about another one.

Let's do another example.

So we're not using this definition anymore.

Again, on complicated mi , mixtures of cosmological parameters.

So what I'd like you to do is fill in these two functions.

Those values are discrete.

So is this a discrete or a continuous random variable?

So if we look over here, so for a T distribution with 9 degrees of freedom, you're going to have 95 of the probability is going to be contained within a T value of so the T value is going to be between negative, so this value right here is 2.262, and this value right here is negative 2.262.

What's the probability of the joint X, Y?

Question 1 In the first question, I'm going to ask you some very basic probability questions.

A probability theorist...

They are not discrete values.

And it makes much more sense to talk about the probability or random variable equaling a value, or the probability that it is less than or greater than something or the probability that is has some property

And they're going to be moving around in this unpredictable way that we would have to describe with a probability function.

You have discrete random variables, and you have continuous random variables.

We're talking about ones that can take on distinct values.

Similarly for tails, if you took a die

And I want to think together about whether you would classify them as discrete or continuous random variables.

Y is the mass of a random animal selected at the New Orleans zoo.

The exact, the precise time that you would see at the men's 100 meter dash.

So the important topics to take away from this segment.

The universe is spacially flat.

Like most thermodynamical quantities, it can be expressed as a function of these two variables.

And in that case the Gaussian distribution, the Gaussian probability density looks

The same thing. A million to one we'd find it.

So, we actually just learned some interesting lessons.

So how do we study properties of clusters as a population?

The duration tells you about either their velocity dispersion or the individual masses, and the total number of lenses you see is telling you something about their density.