# Pmf and cdf relationship test

### Having problems with PMF, CDF and PMF. | Physics Forums

We have already seen the joint CDF for discrete random variables. The joint CDF has the same definition for continuous random variables. It also satisfies the. To compute the cdf at a number of points, we can pass a list or a numpy array. .. However pdf is replaced the probability mass function pmf, no estimation methods, the distributions have been tested over some range of parameters, however in some We see that there is very little difference between Scott's Rule and. For discrete random variables the link between the CDF and the mass . What is the difference between PMF and CDF of a discrete function?.

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.

## Cumulative distribution function

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. 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?

- Relationship between cdf and density (histogram) plots
- Having problems with PMF, CDF and PMF.
- Having problems with PMF, CDF and PMF.

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. Figure c shows the histogram of the signal using the full data set, all k points.

As can be seen, the larger number of samples results in a much smoother appearance. Just as with the mean, the statistical noise roughness of the histogram is inversely proportional to the square root of the number of samples used. From the way it is defined, the sum of all of the values in the histogram must be equal to the number of points in the signal: The histogram can be used to efficiently calculate the mean and standard deviation of very large data sets.

This is especially important for images, which can contain millions of samples. The histogram groups samples together that have the same value.

This allows the statistics to be calculated by working with a few groups, rather than a large number of individual samples. Using this approach, the mean and standard deviation are calculated from the histogram by the equations: Table contains a program for calculating the histogram, mean, and standard deviation using these equations.

Calculation of the histogram is very fast, since it only requires indexing and incrementing.

## Probability density functions

In comparison, calculating the mean and standard deviation requires the time consuming operations of addition and multiplication. The strategy of this algorithm is to use these slow operations only on the few numbers in the histogram, not the many samples in the signal. This makes the algorithm much faster than the previously described methods.

Think a factor of ten for very long signals with the calculations being performed on a general purpose computer. The notion that the acquired signal is a noisy version of the underlying process is very important; so important that some of the concepts are given different names.

### Joint Cumulative Distributive Function| Marginal PMF | CDF

The histogram is what is formed from an acquired signal. The corresponding curve for the underlying process is called the probability mass function pmf. A histogram is always calculated using a finite number of samples, while the pmf is what would be obtained with an infinite number of samples. The pmf can be estimated inferred from the histogram, or it may be deduced by some mathematical technique, such as in the coin flipping example.

Figure shows an example pmf, and one of the possible histograms that could be associated with it. The key to understanding these concepts rests in the units of the vertical axis.

As previously described, the vertical axis of the histogram is the number of times that a particular value occurs in the signal. The vertical axis of the pmf contains similar information, except expressed on a fractional basis.

In other words, each value in the histogram is divided by the total number of samples to approximate the pmf. This means that each value in the pmf must be between zero and one, and that the sum of all of the values in the pmf will be equal to one.

### Having problems with PMF, CDF and PMF. | Physics Forums

The pmf is important because it describes the probability that a certain value will be generated. For example, imagine a signal generated by the process described by Fig. What is the probability that a sample taken from this signal will have a value of ?

Figure b provides the answer, 0. What is the probability that a randomly chosen sample will have a value greater than ? Adding up the values in the pmf for: Thus, the signal would be expected to have a value exceeding on an average of every 82 points. What is the probability that any one sample will be between 0 to ? Summing all of the values in the histogram produces the probability of 1. The histogram and pmf can only be used with discrete data, such as a digitized signal residing in a computer.

A similar concept applies to continuous signals, such as voltages appearing in analog electronics. The probability density function pdfalso called the probability distribution function, is to continuous signals what the probability mass function is to discrete signals.

For example, imagine an analog signal passing through an analog-to-digital converter, resulting in the digitized signal of Fig. For simplicity, we will assume that voltages between 0 and millivolts become digitized into digital numbers between 0 and The pmf of this digital signal is shown by the markers in Fig. Similarly, the pdf of the analog signal is shown by the continuous line in cindicating the signal can take on a continuous range of values, such as the voltage in an electronic circuit.

The vertical axis of the pdf is in units of probability density, rather than just probability.