 # Difference between pdf and cdf graphs of trigonometric functions

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## Graphing Linear Equations Test Review

Even math majors often need a refresher before going into a finance program. This book combines probability, statistics, linear algebra, and multivariable calculus with a view toward finance. In the previous chapter we considered Poisson random variables, for instance the number of earthquakes that occur in two years.

In mathematics , even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis , especially the theory of power series and Fourier series. Evenness and oddness are generally considered for real functions , that is real-valued functions of a real variable. However, the concepts may be more generally defined for functions whose domain and codomain both have a notion of additive inverse. This includes abelian groups , all rings , all fields , and all vector spaces.

### Maths Methods

Mathematical functions play an important role in the GAMS language, especially for nonlinear models. Like other programming languages, GAMS provides a number of built-in or intrinsic functions. GAMS is used in an extremely diverse set of application areas and this creates frequent requests for the addition of new and often sophisticated and specialized functions.

There is a trade-off between satisfying these requests and avoiding complexity not needed by most users. However, these external libraries can currently only provide functionality for the evaluation of functions incl. Solvers that need to analyze the algebraic structure of the model instance are therefore not able to work with extrinsic functions. This includes the class of deterministic global solvers, see column "Global" in this table , while, for example, stochastic global solvers can work with extrinsic functions.

In this chapter we will demonstrate how to access functions from an extrinsic function library in a GAMS model and we will describe the extrinsic function libraries that are included in the GAMS distribution. In addition, we will provide some pointers for users who wish to build their own extrinsic function library. Here InternalLibName is a handle that will be used to refer to the library inside the model source code, ExternalLibName is the file name of the shared library that implements the extrinsic functions.

To access a library that does not reside in these standard places, the external name should include a relative or absolute path to the location of the library. GAMS will then search for the specified library using the mechanisms specific to the host operating system. Before the individual functions may be used, they have to be declared in the following way:.

The user may choose this internal name freely and thus avoid potential naming conflicts. Once functions have been declared in this way they may be used like intrinsic functions. Note that in the first line the external trigonometric library tricclib is activated and the internal name myLib is specified for it.

Then the functions are declared. Observe that Cosine , Sine and Pi are functions in the trigonometric library. After the library has been loaded and the functions have been declared, the functions may be used as usual. The trigonometric library is discussed in section Example: Trigonometric Library below. In the tables that follow, the "Endogenous Classification" last column specifies in which models the function may legally appear.

Note well that functions classified as any are only permitted with exogenous constant arguments. A word on the notation in the tables below: for function arguments, lower case indicates that an endogenous variable is allowed. For details on endogenous variables, see section Functions in Equation Definitions. Upper case function arguments indicate that a constant is required. Arguments in square brackets may be omitted: the default values used in such cases are specified in the function description provided.

Note that the supporting points to which the function will be fit need to be stored in a three-dimensional parameter, fitdata , in a GDX file fit. The first dimension is a function index, the second dimension is the index of the supporting point and the third dimension takes one of the following four values: "w" weight , "x" x-value , "y" y-value or "z" z-value.

It provides the following functions:. The function FitParam may be used to change certain parameters that are used for the evaluation. The following values are defined:. The Piecewise Polynomial Library may be used to evaluate piecewise polynomial functions. Note that the functions that are to be evaluated need to be defined and stored in a GDX file. The following code snippet serves as illustration:. Here FuncInd sets a function index and SegInd defines the index of the segment or interval which is described.

Further, LeftBound gives the lower bound of the segment. The upper bound will be taken from the lower bound on the following segment, or set to infinity in case it is the last segment. Finally, CoefX defines the X -th degree coefficient of the polynomial corresponding to this segment.

It provides the following function:. The Stochastic Library provides random deviates, probability density functions, cumulative density functions and inverse cumulative density functions for certain continuous and discrete distributions.

This library is made available with the following directive:. The continuous distributions that are available with this library are the following:. Note that for each distribution the library offers the following four functions, where DistributionName is the name of the distribution as listed in the tables above, parameters are the parameters associated with each distribution, and x is the point at which the function is to be evaluated. Note that x may be an endogenous variable.

Finally, the seed for the various random number generators can be set by using the following function:. In the following example, a sample of size 20 is generated from the Normal, Binomial, Cauchy, and Lognormal distributions each:. In the example, first the stochastic library is made available in GAMS, then the functions that will be used from the library are declared, giving them names under which to refer to them in the GAMS model. The following tables list the available continuous and discrete distributions, respectively.

The values are 0 meaning "none" , 1 meaning "Latin Hyper Square" and 2 meaning "Antithetic". The default is Latin Hyper Square sampling, it will be used if no variance reduction method is specified. The following example illustrates the use of the sample generator and shows the effect of the functions setCorrelation and induceCorrelation :.

This section discusses the creation of a custom extrinsic function library. Before attempting to implement such a library, we suggest to study the example libraries for which source code and test models are available. These libraries are studied below. An extrinsic function library consists of a specification part and a number of callbacks to evaluate the defined functions at an input point. The specification part is implemented by a callback querylibrary.

It returns information about the library itself, available functions, their arguments, endogenous classification, etc.

C, Delphi, or Fortran source code for this callback can be generated automatically by using the Python helper script ql. The format of this file is documented in the file tri. Both ql. If an extrinsic function will be used within equations of a GAMS model, next to the function value evaluation callback, also callbacks that compute first and second derivatives with respect to all endogenous arguments at an input point should be provided. Occasionally, this can be inconvenient. Observe that GAMS can use the function values at points close to the input point to estimate the derivate values using finite differences.

However, this method is not as accurate as analytic derivatives and requires a number of function evaluations, thus the convenience comes at a price. The attribute MaxDerivative in the specification of a function signals GAMS the highest derivatives this function will provide. For higher order derivatives, GAMS will use finite differences to approximate the derivative values. However, a better alternative is often the use of automatic differentiation when implementing the function evaluation.

GAMS offers some support to check the implementation of of derivatives for extrinsic functions via the function suffixes grad , gradn , hess and hessn. These function suffixes are defined for intrinsic and extrinsic functions. For example, for an extrinsic function userfunc , the gradient evaluation that the user implemented may be called with userfunc.

Further, an approximation of the gradient by finite differences is available by calling userfunc. Comparing the results of these two calls can often help to check the implementation of the gradient. The same principle applies for the Hessian and the function suffixes. The Trigonometric Library serves as an example of how to code and build an extrinsic function library. The library is included in the GAMS distribution in binary form.

The library implements the following extrinsic functions:. The C implementation of this extrinsic function library can be found in the files tricclib. Together with the API specification file extrfunc. The file tricclibql. For example, the information that the function cosine has an endogenous required first argument and an exogenous optional second argument is available from the querylibrary callback.

This library serves as an example of how to code and build an extrinsic function library that reads the information from a GAMS parameter file. The C implementation of this extrinsic function library may be found in the files parcclib.

The API specification file is called par. The LogOption function then only has to return the value of the LogOption parameter from the parameter file. If the option hasn't been specified, the default value will be returned. For further details, see the comments in the file parcclib. These functions are listed and described in the following table. Note that in keeping with the language conventions of statistics, PDF is shorthand for "probability density function" and CDF is shorthand for "cumulative distribution function".

Often, extrinsic functions are created in order to be used with endogenous arguments. In such cases it is necessary to provide first and second derivatives with respect to these arguments in addition to the function values themselves. One way to compute these derivatives is via automatic differentiation techniques see the article in Wikipedia for details. This is the technique used to compute the derivatives in the CPP Library. The source is a working self-documentation of how the process of automatic differentiation works.

We use the standard Normal mean of 0, standard deviation of 1 since intrinsic functions are limited to 20 arguments. The functions for the univariate case are included as convenient examples and should give results nearly identical to the functions pdfNormal and cdfNormal from the stochastic library. For the multivariate cases, the implementation is based on TVPACK from Alan Genz , with some modifications to allow for proper computation of derivatives.

Note that we chose to implement the functions taking correlation coefficients as arguments, not a covariance matrix.

The conversion from a covariance matrix to correlation coefficients is straightforward. The following R code describes this conversion. ## Discrete Vs Continuous

The following are the meanings of the abbreviations and punctuation used in the syntax descriptions in this section. Exp: Expression Value, Variable, etc. Rounds the decimal part of a value to the specified number of decimal places. Rounds a value to the specified number of digits. Specifying two integer values for the argument generates random numbers between them. You can specify an integer from 0 to 9 for the argument of this command.

Mathematical functions play an important role in the GAMS language, especially for nonlinear models. Like other programming languages, GAMS provides a number of built-in or intrinsic functions. GAMS is used in an extremely diverse set of application areas and this creates frequent requests for the addition of new and often sophisticated and specialized functions. There is a trade-off between satisfying these requests and avoiding complexity not needed by most users. However, these external libraries can currently only provide functionality for the evaluation of functions incl. Solvers that need to analyze the algebraic structure of the model instance are therefore not able to work with extrinsic functions.

In probability theory and statistics , a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. For instance, if X is used to denote the outcome of a coin toss "the experiment" , then the probability distribution of X would take the value 0. Examples of random phenomena include the weather condition in a future date, the height of a person, the fraction of male students in a school, the results of a survey , etc. A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. To define probability distributions for the specific case of random variables so the sample space can be seen as a numeric set , it is common to distinguish between discrete and continuous random variables. The probability of an event is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event "the dice rolls an even value" is. Probability Density Function Problem? Thats exponential distribution you are talking about, just like Normal Distribution, Uniform Distribution. See here to see more detail on it. How will i determine then the offset of sine wave to ensure that my sinewave is in the center of range. How can probability be infinity for -1 and 1, the pdf has extreme values there? You wish to use a parametric probability distribution that is not provided by ModelRisk , and you know:. The cumulative distribution function continuous variable ;.

### Graphing Linear Equations Test Review

Did you know that finding the probability of a continuous random variable is nothing more than using integration? A discrete random variable is a one that can take on a finite or countable infinite sequence of elements as noted by the University of Florida. In contrast, a continuous random variable is a one that can take on any value of a specified domain i. For example, the height of students in a class, the amount of ice tea in a glass, the change in temperature throughout a day, and the number of hours a person works in a week all contain a range of values in an interval, thus continuous random variables. What is important to note is that discrete random variables use a probability mass function PMF but for continuous random variables, we say it is a probability density function PDF , or just density function. So, to find the probability, we just need to integrate over the region using our knowledge of the Fundamental Theorem of Calculus! We define a cumulative density function CDF to calculate the area under the curve in these instances.

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