probability - Do the pdf and the pmf and the cdf contain the same information? - Cross Validated
Thus, the interpretation of the CDF is the same whether we have a discrete or continuous variable (read pdf or pmf), but the definition is slightly different. There is a theorem called the Riemann Rearrangement Theorem that says that your rearragement of the infinite series is allowed (since it is absolutely. To begin looking at PMF, PDF, and CDF, we must first identify our sample space. It is important to note that the difference between discrete and continuous sets The probabilities with associated x values can be represented in many ways.
Outcomes for flipping a coin are heads or tails The number of gallons of milk on a shelf that has a max capacity of There are 21 possible outcomes: Examples of a Continuous Set: The exact heights of a group students in a class.
The exact amount of milk in a cup. These examples may sound similar, but the measurements of interest are not. The key difference is how many times we can break up the unit we are using. For instance, no human has ever been under a 1 foot, or over 12 feet tall.Finding a CDF from a pdf
It seems logical that measuring height would be discrete, but the example above asked for exact height. There are very few people in the world that are exactly 6 feet tall; however, there are people who are 6.
There infinitely many possibilities between 6 feet and 7 feet, let alone the rest of the sample space.
Relationship between cdf and density (histogram) plots
However, if the question asked for the heights in whole feet or inches, the sample space would be a discrete set. It is important to note that the difference between discrete and continuous sets is not that one has infinitely many possibilities and the other does not.
There are discrete sets that have essentially infinity possible outcomes too. For example, if we counted the number of whole grains of sand on a beach or the world. Practically, this is impossible to count, but there is a finite amount of possibilities.
Random Variables Now that we have talked about sample spaces and the potential outcomes of sample spaces, the next topic is the probability of these outcomes. The first property of interest is that the sum of all probabilities within a sample space must be less than or equal to 1.
This may seem arbitrary, but it is actually far from random. This property allows us to interpret these probabilities the same way as we interpret percentages. Let's go back to the coin flip example above, we had two possible outcomes: Logically it would make sense that probability of a specific number of outcomes would be one divided by the number outcomes in the sample space N. For the coin flip example this would be correct, the probability of each outcome is.
We will represent the histogram by Hi, where i is an index that runs from 0 to M-1, and M is the number of possible values that each sample can take on. For instance, H50 is the number of samples that have a value of 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.
probability - Relationship between pmf and cdf - Mathematics Stack Exchange
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. 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.
PMF, PDF, CDF - StatsQueen
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. 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. 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.