Probability Density Function - Research Paper Example

Abstract

The probability density function tends to determine the distribution of the discrete random variable. This is well applicable when finding out the CDF or PMF of the discrete random variable. For random variables that are continuous, CDF is properly defined so as we could provide the CDF. However, PMF is not applicable for the continuous random variables since for the continuous random variable P (X=x) is always perceived to be 0. It is at this point that probability density is found applicable. In determining the distribution in the cases of random variables, there are two options to be used. The CDF or PMF of the random variables can be determined. For random variables that are continuous, those variables applies CDF since they are well defined within their continuous limits. PMF does not, however, work for the continuous random variables, probability density function can be defined as the density of the probability and not the probability mass (Epanechnikov, 2009). The concept of Probability Density Function is similar to that used in mass density in physics. Its unit is considered the probability per unit length. PDF is considered a continuous variable and a function where any value given to it a given sample is the set of possibilities that can be taken by the random variable (Epanechnikov, 2009). This can be interpreted that the likelihood for a random value to be equal to any particular value is zero since there are infinite values that can be considered.

Probability Density Function

The probability density function can well be defined as the density of probability instead of probability mass. The concept of probability density function is slightly similar to mass densities with a standard unit of probability per unit length. To properly define the PDF it is therefore important to consider random variables that are continuous to be a function of fX(x). The function fX(x) provides us with the probability density at a point x. There is a situation in the use of probability density function that has been confusing. In some sources, PDF can be applied in the situations where the probability function is considered as a function over the general value sets. In these cases, it may be referred to as probability mass function or cumulative distribution function rather than using the term density (Al-Habash, Andrews, & Phillips, 2001). In the model, density function can also be used for probability mass function. This leads to more confusion. In these cases though the probability mass function is applied to random variables, the probability density function is used in cases of continuous random variables.

Probability Density Function

The probability density function of a random variable x enables one to compute the probability of an event. For increase, for distributions that are continuous, the probability that the continuous variable has values within a defined interval, is precisely the region under its probability density function in the defined interval. For the distributions that are discrete, the probability that the discrete variable has values in a defined interval, the values are the summation of probability density function of the possible discrete values of the discrete variable in the interval.

Literature Review

Random variables are defined to be a collection of values with varying probabilities. Such random variables can either be discrete or continuous (Al-Habash et al., 2001). Discrete variables have a finite number of values and outcomes while the continuous variable has infinite possibilities of values. For instance, the discrete values could be the number of dead and those that are alive, those who have passed and those who have failed, dice, and also counts. Infinite possibilities of values include the height of an individual, the weight of a person, the speedometer, and other real numbers. It is important in the mathematical background to explain the idea of probability density function. According to Fukunaga and Hostetler (2005), when a number is picked, let us say x, between the values 0 and 10 and nothing is explained, there is the probability that x can be any number in that range. Assuming there is no preference for any of the numbers, then the probability of having any number remains the same. Since the probability has to be made to add up to 1, then the logical conclusion is that the probability of each integer will be. It is assumed here that the probability that x is one in and this applies to the other integers between 0 and 10. There will be change in cases where the number x is between 0 and 1. It is assumed that there are only two possibilities, the number being 1 or 0 and a probability of may be assigned to each.

Definition

The probability density function can well be defined as a function which describes the relative likelihood of a random variable to take on any form of value. The function is provided by the integral of variable densities over a certain range. The probability density function could be represented by the region below the density function and above the horizontal axis and also between the greatest and the lowest values within a certain range. The definition of PDF can be given in cases of continuous variables where the reference measure in these cases is the Lebesgue measure. The probability mass function when discrete random variables are considered is the density considering the counting measure of the set of integers that act as the sample space. In these definitions, the density cannot be defined with the reference made to arbitrary measure (Fukunaga & Hostetler, 2005). In the definition of PDF, there cannot be a choice to the counting measure to be used a reference for a continuous random variable. The density is everywhere unit in cases the continuous random variable does not exist.

Application

In the application of PDF, they can be used and take a value that is greater than one. This considered whether the distribution is standard formal or uniform distribution. It is also important to consider that not all the probability functions can be assigned a density function. Cantor distribution and discrete random distribution do kb have the density function even though they do no discrete point. It implies that the distribution does not assign any positive probability to any given point. The individual points are not assigned any density. The density function in a distributor exists only in the cases where the cumulative functions of the distributor are absolutely continuous. This can be presented as F (x). F in this function is differentiable and everywhere and its derivative can be applied and used in other functions as probability density.

In cases when a density is admitted in a probability distribution, then every single point has a probability of zero (0). The same principle is applied for countable and finite sets. When two probability densities are considered; let us say f and g, they represent the same probability distribution and they are expected to differ only in the cases of Labesgue measure zero. The principle has been applied in other areas such as physics. Statistical physics has had a non-formal reformulation on the link above between the probability density function and cumulative distribution function which is generally used in the definition of probability density functions.

In the definition of probability density function, it is important to look at the link between continuous and discrete distributions. In the calculations, it is possible to represent some random variable and discrete random variables that are in involved in both discrete and continuous parts through a generalized density function. The representation can be done through the use of Dirac delta function. In an example, when a binary discrete random variable is considered to have a Rademarcher distribution, -1 or 1 can be taken as the values with the probability of each being 0.5. The density of probability that might be obtained in this case is:

F (t) = (

In general terms, in cases where a discrete variable can be made to take a value n among other real numbers, the associated probability density function can be obtained as shown below.

F (t) = where ……. are the discrete values that are accessible to the variables and ……. are the probabilities that are associated with the values.

The substitution in the case above unites the treatment of continuous and discrete probability distributions. In an example, the expression given above enables the determination of mean, variance and other statistical characteristics of the discrete variable. The determination is through the formula given for distribution of the probability.

Mathematical Background

Assuming that the F (x) is a random variable that is continuous,

f (x) = dF (x) / dx

the CDF would be

Formulating the function would bring forth various mathematical formula for the function.

From the above properties, the first one property applies conventional calculus theories. The second one shows that PDF function is always perceived to be non-negative. The issue makes sense because probabilities are normally non-negative values. From the previous formulas, the CDF of the function F (x) is normally assumed to be the non-decreasing function of the value c. Therefore, the derivation of the function f(x) is assumed to be non-negative. The third property shows that the region between the x-axis and the function has to be one. This is true because

Real-Life application

One of the applications of probability density function is in research when testing for the hypothesis. When researchers want to determine whether the results they obtained from a study are statistically significant, they have to establish the relationship between two quantities. In these establishments, they will end up integrating a Gaussian density as some points of their calculation. Quantum mechanics is another area where the probability density function is applied. Quantum mechanics has applied the model in explaining a wide range of physical experiments. There are several examples where it has been used in this field. In an example, the model can be used to describe the time that an isotope may take to decay. In the calculation of normal distributions, the model can also be applied in determining the momentum vector of a molecule when it is in a gas (Parzen, 2012). Radioactive decay can be monitored through this model knowing that the radioactive decay takes some period of time. The interval can be determined.

Conclusion

The probability distribution function of a random variable tends to describe the manner in which probability distribution functions are aligned among random values. The function is within the PDF limits which provides the probability of numbers within both the discrete and the continuous random variables. The continuous values assumes any given value within the sets of intervals. It is therefore, impossible talking on the probability of a random variable keeping in mind those values within the defined interval. The probability of those variables tends to assume a range of values within an interval which are normally portrayed below the graphical functions of two values. In the applications of probability density functions, it is important to consider the range of the integers and the values that are being dealt with. The range of a random variable in these cases is the part of possibilities that can be obtained in a random variable. For instance, in situations where x is considered to be a random variable that is on a continuous trend, the range is signified by set of values where the probability density function obtained is more than zero.

Summary

When one applies the probability density functions, it is important to take into consideration the densities examined. Some of the families include marginal densities, independent densities, and corollary densities. Real life has applied this model in several cases and any researcher is expected to have the mathematical background and the cases where the model can be applied. Probability theory is applied in almost any imaginable situations. It is hence important to start believing that the real life and most of the activities we carry out are mainly governed by probability. Many of the activities are based on hypothesis and in determining the possible outcomes in such hypothesis, probability density function has been applied. It is evident that in the cases where x is allowed to be real numbers, this will be a wring approach. There is no possibility that x is just a single number. The probability is zero. It should be thought that x is close to a single number. The notion of being a close to a number is captured with the probability density function. Since PDF is considered the derivative of CDF, then CDF can be obtained from PDF through integration. In particular, when the integration of PDF is done over the entire line, then the value that is obtained must be 1. This implies that the area under the curve when calculated must be equal to one.

Probability Density Function

The probability density function of a random variable x enables one to compute the probability of an event. For increase, for distributions that are continuous, the probability that the continuous variable has values within a defined interval, is precisely the region under its probability density function in the defined interval. For the distributions that are discrete, the probability that the discrete variable has values in a defined interval, the values are the summation of probability density function of the possible discrete values of the discrete variable in the interval.

Literature Review

Random variables are defined to be a collection of values with varying probabilities. Such random variables can either be discrete or continuous (Al-Habash et al., 2001). Discrete variables have a finite number of values and outcomes while the continuous variable has infinite possibilities of values. For instance, the discrete values could be the number of dead and those that are alive, those who have passed and those who have failed, dice, and also counts. Infinite possibilities of values include the height of an individual, the weight of a person, the speedometer, and other real numbers. It is important in the mathematical background to explain the idea of probability density function. According to Fukunaga and Hostetler (2005), when a number is picked, let us say x, between the values 0 and 10 and nothing is explained, there is the probability that x can be any number in that range. Assuming there is no preference for any of the numbers, then the probability of having any number remains the same. Since the probability has to be made to add up to 1, then the logical conclusion is that the probability of each integer will be. It is assumed here that the probability that x is one in and this applies to the other integers between 0 and 10. There will be change in cases where the number x is between 0 and 1. It is assumed that there are only two possibilities, the number being 1 or 0 and a probability of may be assigned to each.

Definition

The probability density function can well be defined as a function which describes the relative likelihood of a random variable to take on any form of value. The function is provided by the integral of variable densities over a certain range. The probability density function could be represented by the region below the density function and above the horizontal axis and also between the greatest and the lowest values within a certain range. The definition of PDF can be given in cases of continuous variables where the reference measure in these cases is the Lebesgue measure. The probability mass function when discrete random variables are considered is the density considering the counting measure of the set of integers that act as the sample space. In these definitions, the density cannot be defined with the reference made to arbitrary measure (Fukunaga & Hostetler, 2005). In the definition of PDF, there cannot be a choice to the counting measure to be used a reference for a continuous random variable. The density is everywhere unit in cases the continuous random variable does not exist.

Application

In the application of PDF, they can be used and take a value that is greater than one. This considered whether the distribution is standard formal or uniform distribution. It is also important to consider that not all the probability functions can be assigned a density function. Cantor distribution and discrete random distribution do kb have the density function even though they do no discrete point. It implies that the distribution does not assign any positive probability to any given point. The individual points are not assigned any density. The density function in a distributor exists only in the cases where the cumulative functions of the distributor are absolutely continuous. This can be presented as F (x). F in this function is differentiable and everywhere and its derivative can be applied and used in other functions as probability density. Read More