# Pdf And Cdf Of All Distributions Explained

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Published: 08.06.2021  Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. It only takes a minute to sign up. So, in this sense, the CDF is indeed as fundamental as the distribution itself.

## What is Probability Density Function (PDF)?

Math Statistics and probability Random variables Continuous random variables. Probability density functions. Probabilities from density curves. Practice: Probability in density curves. Practice: Probability in normal density curves. Next lesson. Current timeTotal duration Google Classroom Facebook Twitter. Video transcript In the last video, I introduced you to the notion of-- well, really we started with the random variable.

And then we moved on to the two types of random variables. You had discrete, that took on a finite number of values. And the these, I was going to say that they tend to be integers, but they don't always have to be integers. You have discrete, so finite meaning you can't have an infinite number of values for a discrete random variable. And then we have the continuous, which can take on an infinite number.

And the example I gave for continuous is, let's say random variable x. And people do tend to use-- let me change it a little bit, just so you can see it can be something other than an x. Let's have the random variable capital Y. They do tend to be capital letters.

Is equal to the exact amount of rain tomorrow. And I say rain because I'm in northern California. It's actually raining quite hard right now. We're short right now, so that's a positive. We've been having a drought, so that's a good thing. But the exact amount of rain tomorrow. And let's say I don't know what the actual probability distribution function for this is, but I'll draw one and then we'll interpret it.

Just so you can kind of think about how you can think about continuous random variables. So let me draw a probability distribution, or they call it its probability density function.

And we draw like this. And let's say that there is-- it looks something like this. Like that. All right, and then I don't know what this height is. So the x-axis here is the amount of rain. Where this is 0 inches, this is 1 inch, this is 2 inches, this is 3 inches, 4 inches.

And then this is some height. Let's say it peaks out here at, I don't know, let's say this 0. So the way to think about it, if you were to look at this and I were to ask you, what is the probability that Y-- because that's our random variable-- that Y is exactly equal to 2 inches? That Y is exactly equal to two inches. What's the probability of that happening? Well, based on how we thought about the probability distribution functions for the discrete random variable, you'd say OK, let's see.

Let me go up here. You'd say it looks like it's about 0. And you'd say, I don't know, is it a 0. And I would say no, it is not a 0. And before we even think about how we would interpret it visually, let's just think about it logically.

What is the probability that tomorrow we have exactly 2 inches of rain? Not 2. Not 1. Exactly 2 inches of rain. I mean, there's not a single extra atom, water molecule above the 2 inch mark. And not as single water molecule below the 2 inch mark. It's essentially 0, right? It might not be obvious to you, because you've probably heard, oh, we had 2 inches of rain last night. But think about it, exactly 2 inches, right?

Normally if it's 2. But we're saying no, this does not count. It can't be 2 inches. We want exactly 2. Normally our measurements, we don't even have tools that can tell us whether it is exactly 2 inches.

No ruler you can even say is exactly 2 inches long. At some point, just the way we manufacture things, there's going to be an extra atom on it here or there. So the odds of actually anything being exactly a certain measurement to the exact infinite decimal point is actually 0. The way you would think about a continuous random variable, you could say what is the probability that Y is almost 2?

So if we said that the absolute value of Y minus is 2 is less than some tolerance? Is less than 0. And if that doesn't make sense to you, this is essentially just saying what is the probability that Y is greater than 1. These two statements are equivalent. I'll let you think about it a little bit. But now this starts to make a little bit of sense. Now we have an interval here. So we want all Y's between 1. So we are now talking about this whole area.

And area is key. So if you want to know the probability of this occurring, you actually want the area under this curve from this point to this point. And for those of you who have studied your calculus, that would essentially be the definite integral of this probability density function from this point to this point.

So from-- let me see, I've run out of space down here. So let's say if this graph-- let me draw it in a different color. If this line was defined by, I'll call it f of x. I could call it p of x or something. The probability of this happening would be equal to the integral, for those of you who've studied calculus, from 1. Assuming this is the x-axis. So it's a very important thing to realize. Because when a random variable can take on an infinite number of values, or it can take on any value between an interval, to get an exact value, to get exactly 1.

It's like asking you what is the area under a curve on just this line. Or even more specifically, it's like asking you what's the area of a line? An area of a line, if you were to just draw a line, you'd say well, area is height times base. Well the height has some dimension, but the base, what's the width the a line?

As far as the way we've defined a line, a line has no with, and therefore no area. And it should make intuitive sense. That the probability of a very super-exact thing happening is pretty much 0.

That you really have to say, OK what's the probably that we'll get close to 2? And then you can define an area. And if you said oh, what's the probability that we get someplace between 1 and 3 inches of rain, then of course the probability is much higher. The probability is much higher. It would be all of this kind of stuff. You could also say what's the probability we have less than 0.

Then you would go here and if this was 0. Say you were to take a coin from your pocket and toss it into the air. While it flips through space, what could you possibly say about its future? Will it land heads up? More than that, how long will it remain in the air? How many times will it bounce?

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You might recall that the cumulative distribution function is defined for discrete random variables as:. The cumulative distribution function for continuous random variables is just a straightforward extension of that of the discrete case. All we need to do is replace the summation with an integral. The cumulative distribution function " c. Now for the other two intervals:. Therefore, the graph of the cumulative distribution function looks something like this:. Breadcrumb Home 14 Probability Density Functions (PDF):​​ A PDF is simply the derivative of a CDF. Thus a PDF is also a function of a random variable, x, and its magnitude will be some indication of the relative likelihood of measuring a particular value.

## Basic Statistical Background

### Probability density functions

Cumulative distribution functions are also used to specify the distribution of multivariate random variables. The proper use of tables of the binomial and Poisson distributions depends upon this convention. The probability density function of a continuous random variable can be determined from the cumulative distribution function by differentiating  using the Fundamental Theorem of Calculus ; i. Every function with these four properties is a CDF, i. Sometimes, it is useful to study the opposite question and ask how often the random variable is above a particular level. This is called the complementary cumulative distribution function ccdf or simply the tail distribution or exceedance , and is defined as.

Chapter 2: Basic Statistical Background. Generate Reference Book: File may be more up-to-date. This section provides a brief elementary introduction to the most common and fundamental statistical equations and definitions used in reliability engineering and life data analysis. In general, most problems in reliability engineering deal with quantitative measures, such as the time-to-failure of a component, or qualitative measures, such as whether a component is defective or non-defective. Our component can be found failed at any time after time 0 e. In this reference, we will deal almost exclusively with continuous random variables. The cumulative distribution function F(x) for a continuous rv X is defined for every number x by. F(x) = P(X ≤ x) = For each x, F(x) is the area under the density. Exploratory Data Analysis 1. EDA Techniques 1. Probability Distributions 1. Probability distributions are typically defined in terms of the probability density function. However, there are a number of probability functions used in applications.

In probability theory , a probability density function PDF , or density of a continuous random variable , is a function whose value at any given sample or point in the sample space the set of possible values taken by the random variable can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample. In a more precise sense, the PDF is used to specify the probability of the random variable falling within a particular range of values , as opposed to taking on any one value. This probability is given by the integral of this variable's PDF over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to 1.

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