6 Probability Density Functions (Pdfs)
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Problem Set 2
22.02 – Introduction to Applied Nuclear Physics Problem set # 2 Issued on Wednesday Feb. 22, 2012. Due on Wednesday Feb. 29, 2012 Problem 1: Gaussian Integral (solved problem) 2 2 1 x /2σ The Gaussian function g(x) = e− is often used to describe the shape of a wave packet. Also, it represents √2πσ2 the probability density function (p.d.f.) of the Gaussian distribution. 2 2 ∞ 1 x /2σ a) Calculate the integral √ 2 e− −∞ 2πσ Solution R 2 2 x x Here I will give the calculation for the simpler function: G(x) = e− . The integral I = ∞ e− can be squared as: R−∞ 2 2 2 2 2 2 ∞ x ∞ ∞ x y ∞ ∞ (x +y ) I = dx e− = dx dy e− e− = dx dy e− Z Z Z Z Z −∞ −∞ −∞ −∞ −∞ This corresponds to making an integral over a 2D plane, defined by the cartesian coordinates x and y. We can perform the same integral by a change of variables to polar coordinates: x = r cos ϑ y = r sin ϑ Then dxdy = rdrdϑ and the integral is: 2π 2 2 2 ∞ r ∞ r I = dϑ dr r e− = 2π dr r e− Z0 Z0 Z0 Now with another change of variables: s = r2, 2rdr = ds, we have: 2 ∞ s I = π ds e− = π Z0 2 x Thus we obtained I = ∞ e− = √π and going back to the function g(x) we see that its integral just gives −∞ ∞ g(x) = 1 (as neededR for a p.d.f). −∞ 2 2 R (x+b) /c Note: we can generalize this result to ∞ ae− dx = ac√π R−∞ Problem 2: Fourier Transform Give the Fourier transform of : (a – solved problem) The sine function sin(ax) Solution The Fourier Transform is given by: [f(x)][k] = 1 ∞ dx e ikxf(x). -
The Error Function Mathematical Physics
R. I. Badran The Error Function Mathematical Physics The Error Function and Stirling’s Formula The Error Function: x 2 The curve of the Gaussian function y e is called the bell-shaped graph. The error function is defined as the area under part of this curve: x 2 2 erf (x) et dt 1. . 0 There are other definitions of error functions. These are closely related integrals to the above one. 2. a) The normal or Gaussian distribution function. x t2 1 1 1 x P(, x) e 2 dt erf ( ) 2 2 2 2 Proof: Put t 2u and proceed, you might reach a step of x 1 2 P(0, x) eu du P(,x) P(,0) P(0,x) , where 0 1 x P(0, x) erf ( ) Here you can prove that 2 2 . This can be done by using the definition of error function in (1). 0 u2 I I e du Now you need to find P(,0) where . To find this integral you have to put u=x first, then u= y and multiply the two resulting integrals. Make the change of variables to polar coordinate you get R. I. Badran The Error Function Mathematical Physics 0 2 2 I 2 er rdr d 0 From this latter integral you get 1 I P(,0) 2 and 2 . 1 1 x P(, x) erf ( ) 2 2 2 Q. E. D. x 2 t 1 2 1 x 2.b P(0, x) e dt erf ( ) 2 0 2 2 (as proved earlier in 2.a). -
Ph 21.5: Covariance and Principal Component Analysis (PCA)
Ph 21.5: Covariance and Principal Component Analysis (PCA) -v20150527- Introduction Suppose we make a measurement for which each data sample consists of two measured quantities. A simple example would be temperature (T ) and pressure (P ) taken at time (t) at constant volume(V ). The data set is Ti;Pi ti N , which represents a set of N measurements. We wish to make sense of the data and determine thef dependencej g of, say, P on T . Suppose P and T were for some reason independent of each other; then the two variables would be uncorrelated. (Of course we are well aware that P and V are correlated and we know the ideal gas law: PV = nRT ). How might we infer the correlation from the data? The tools for quantifying correlations between random variables is the covariance. For two real-valued random variables (X; Y ), the covariance is defined as (under certain rather non-restrictive assumptions): Cov(X; Y ) σ2 (X X )(Y Y ) ≡ XY ≡ h − h i − h i i where ::: denotes the expectation (average) value of the quantity in brackets. For the case of P and T , we haveh i Cov(P; T ) = (P P )(T T ) h − h i − h i i = P T P T h × i − h i × h i N−1 ! N−1 ! N−1 ! 1 X 1 X 1 X = PiTi Pi Ti N − N N i=0 i=0 i=0 The extension of this to real-valued random vectors (X;~ Y~ ) is straighforward: D E Cov(X;~ Y~ ) σ2 (X~ < X~ >)(Y~ < Y~ >)T ≡ X~ Y~ ≡ − − This is a matrix, resulting from the product of a one vector and the transpose of another vector, where X~ T denotes the transpose of X~ . -
Neural Network for the Fast Gaussian Distribution Test Author(S)
Document Title: Neural Network for the Fast Gaussian Distribution Test Author(s): Igor Belic and Aleksander Pur Document No.: 208039 Date Received: December 2004 This paper appears in Policing in Central and Eastern Europe: Dilemmas of Contemporary Criminal Justice, edited by Gorazd Mesko, Milan Pagon, and Bojan Dobovsek, and published by the Faculty of Criminal Justice, University of Maribor, Slovenia. This report has not been published by the U.S. Department of Justice. To provide better customer service, NCJRS has made this final report available electronically in addition to NCJRS Library hard-copy format. Opinions and/or reference to any specific commercial products, processes, or services by trade name, trademark, manufacturer, or otherwise do not constitute or imply endorsement, recommendation, or favoring by the U.S. Government. Translation and editing were the responsibility of the source of the reports, and not of the U.S. Department of Justice, NCJRS, or any other affiliated bodies. IGOR BELI^, ALEKSANDER PUR NEURAL NETWORK FOR THE FAST GAUSSIAN DISTRIBUTION TEST There are several problems where it is very important to know whether the tested data are distributed according to the Gaussian law. At the detection of the hidden information within the digitized pictures (stega- nography), one of the key factors is the analysis of the noise contained in the picture. The incorporated noise should show the typically Gaussian distribution. The departure from the Gaussian distribution might be the first hint that the picture has been changed – possibly new information has been inserted. In such cases the fast Gaussian distribution test is a very valuable tool. -
Covariance of Cross-Correlations: Towards Efficient Measures for Large-Scale Structure
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by RERO DOC Digital Library Mon. Not. R. Astron. Soc. 400, 851–865 (2009) doi:10.1111/j.1365-2966.2009.15490.x Covariance of cross-correlations: towards efficient measures for large-scale structure Robert E. Smith Institute for Theoretical Physics, University of Zurich, Zurich CH 8037, Switzerland Accepted 2009 August 4. Received 2009 July 17; in original form 2009 June 13 ABSTRACT We study the covariance of the cross-power spectrum of different tracers for the large-scale structure. We develop the counts-in-cells framework for the multitracer approach, and use this to derive expressions for the full non-Gaussian covariance matrix. We show that for the usual autopower statistic, besides the off-diagonal covariance generated through gravitational mode- coupling, the discreteness of the tracers and their associated sampling distribution can generate strong off-diagonal covariance, and that this becomes the dominant source of covariance as spatial frequencies become larger than the fundamental mode of the survey volume. On comparison with the derived expressions for the cross-power covariance, we show that the off-diagonal terms can be suppressed, if one cross-correlates a high tracer-density sample with a low one. Taking the effective estimator efficiency to be proportional to the signal-to-noise ratio (S/N), we show that, to probe clustering as a function of physical properties of the sample, i.e. cluster mass or galaxy luminosity, the cross-power approach can outperform the autopower one by factors of a few. -
Lecture 4 Multivariate Normal Distribution and Multivariate CLT
Lecture 4 Multivariate normal distribution and multivariate CLT. T We start with several simple observations. If X = (x1; : : : ; xk) is a k 1 random vector then its expectation is × T EX = (Ex1; : : : ; Exk) and its covariance matrix is Cov(X) = E(X EX)(X EX)T : − − Notice that a covariance matrix is always symmetric Cov(X)T = Cov(X) and nonnegative definite, i.e. for any k 1 vector a, × a T Cov(X)a = Ea T (X EX)(X EX)T a T = E a T (X EX) 2 0: − − j − j � We will often use that for any vector X its squared length can be written as X 2 = XT X: If we multiply a random k 1 vector X by a n k matrix A then the covariancej j of Y = AX is a n n matrix × × × Cov(Y ) = EA(X EX)(X EX)T AT = ACov(X)AT : − − T Multivariate normal distribution. Let us consider a k 1 vector g = (g1; : : : ; gk) of i.i.d. standard normal random variables. The covariance of g is,× obviously, a k k identity × matrix, Cov(g) = I: Given a n k matrix A, the covariance of Ag is a n n matrix × × � := Cov(Ag) = AIAT = AAT : Definition. The distribution of a vector Ag is called a (multivariate) normal distribution with covariance � and is denoted N(0; �): One can also shift this disrtibution, the distribution of Ag + a is called a normal distri bution with mean a and covariance � and is denoted N(a; �): There is one potential problem 23 with the above definition - we assume that the distribution depends only on covariance ma trix � and does not depend on the construction, i.e. -
The Variance Ellipse
The Variance Ellipse ✧ 1 / 28 The Variance Ellipse For bivariate data, like velocity, the variabililty can be spread out in not one but two dimensions. In this case, the variance is now a matrix, and the spread of the data is characterized by an ellipse. This variance ellipse eccentricity indicates the extent to which the variability is anisotropic or directional, and the orientation tells the direction in which the variability is concentrated. ✧ 2 / 28 Variance Ellipse Example Variance ellipses are a very useful way to analyze velocity data. This example compares velocities observed by a mooring array in Fram Strait with velocities in two numerical models. From Hattermann et al. (2016), “Eddydriven recirculation of Atlantic Water in Fram Strait”, Geophysical Research Letters. Variance ellipses can be powerfully combined with lowpassing and bandpassing to reveal the geometric structure of variability in different frequency bands. ✧ 3 / 28 Understanding Ellipses This section will focus on understanding the properties of the variance ellipse. To do this, it is not really possible to avoid matrix algebra. Therefore we will first review some relevant mathematical background. ✧ 4 / 28 Review: Rotations T The most important action on a vector z ≡ [u v] is a ninety-degree rotation. This is carried out through the matrix multiplication 0 −1 0 −1 u −v z = = . [ 1 0 ] [ 1 0 ] [ v ] [ u ] Note the mathematically positive direction is counterclockwise. A general rotation is carried out by the rotation matrix cos θ − sin θ J(θ) ≡ [ sin θ cos θ ] cos θ − sin θ u u cos θ − v sin θ J(θ) z = = . -
Error and Complementary Error Functions Outline
Error and Complementary Error Functions Reading Problems Outline Background ...................................................................2 Definitions .....................................................................4 Theory .........................................................................6 Gaussian function .......................................................6 Error function ...........................................................8 Complementary Error function .......................................10 Relations and Selected Values of Error Functions ........................12 Numerical Computation of Error Functions ..............................19 Rationale Approximations of Error Functions ............................21 Assigned Problems ..........................................................23 References ....................................................................27 1 Background The error function and the complementary error function are important special functions which appear in the solutions of diffusion problems in heat, mass and momentum transfer, probability theory, the theory of errors and various branches of mathematical physics. It is interesting to note that there is a direct connection between the error function and the Gaussian function and the normalized Gaussian function that we know as the \bell curve". The Gaussian function is given as G(x) = Ae−x2=(2σ2) where σ is the standard deviation and A is a constant. The Gaussian function can be normalized so that the accumulated area under the -
Principal Components Analysis (Pca)
PRINCIPAL COMPONENTS ANALYSIS (PCA) Steven M. Holand Department of Geology, University of Georgia, Athens, GA 30602-2501 3 December 2019 Introduction Suppose we had measured two variables, length and width, and plotted them as shown below. Both variables have approximately the same variance and they are highly correlated with one another. We could pass a vector through the long axis of the cloud of points and a second vec- tor at right angles to the first, with both vectors passing through the centroid of the data. Once we have made these vectors, we could find the coordinates of all of the data points rela- tive to these two perpendicular vectors and re-plot the data, as shown here (both of these figures are from Swan and Sandilands, 1995). In this new reference frame, note that variance is greater along axis 1 than it is on axis 2. Also note that the spatial relationships of the points are unchanged; this process has merely rotat- ed the data. Finally, note that our new vectors, or axes, are uncorrelated. By performing such a rotation, the new axes might have particular explanations. In this case, axis 1 could be regard- ed as a size measure, with samples on the left having both small length and width and samples on the right having large length and width. Axis 2 could be regarded as a measure of shape, with samples at any axis 1 position (that is, of a given size) having different length to width ratios. PC axes will generally not coincide exactly with any of the original variables. -
3 Random Vectors
10 3 Random Vectors X1 . 3.1 Definition: A random vector is a vector of random variables X = . . Xn E[X1] . 3.2 Definition: The mean or expectation of X is defined as E[X]= . . E[Xn] 3.3 Definition: A random matrix is a matrix of random variables Z =(Zij). Its expectation is given by E[Z]=(E[Zij ]). 3.4 Theorem: A constant vector a (vector of constants) and a constant matrix A (matrix of constants) satisfy E[a]=a and E[A]=A. 3.5 Theorem: E[X + Y]=E[X]+E[Y]. 3.6 Theorem: E[AX]=AE[X] for a constant matrix A. 3.7 Theorem: E[AZB + C]=AE[Z]B + C if A, B, C are constant matrices. 3.8 Definition: If X is a random vector, the covariance matrix of X is defined as var(X )cov(X ,X ) cov(X ,X ) 1 1 2 ··· 1 n cov(X2,X1)var(X2) cov(X2,Xn) cov(X) [cov(X ,X )] ··· . i j . .. ≡ ≡ . cov(X ,X )cov(X ,X ) var(X ) n 1 n 2 ··· n An alternative form is X1 E[X1] − . cov(X)=E[(X E[X])(X E[X])!]=E . (X E[X ], ,X E[X ]) . − − . 1 − 1 ··· n − n Xn E[Xn] − 3.9 Example: If X1,...,Xn are independent, then the covariances are 0 and the covariance matrix is equal to 2 2 2 2 diag(σ1,...,σn) ,or σ In if the Xi have common variance σ . Properties of covariance matrices: 3.10 Theorem: Symmetry: cov(X)=[cov(X)]!. -
Covariance Matrix Estimation in Time Series Wei Biao Wu and Han Xiao June 15, 2011
Covariance Matrix Estimation in Time Series Wei Biao Wu and Han Xiao June 15, 2011 Abstract Covariances play a fundamental role in the theory of time series and they are critical quantities that are needed in both spectral and time domain analysis. Es- timation of covariance matrices is needed in the construction of confidence regions for unknown parameters, hypothesis testing, principal component analysis, predic- tion, discriminant analysis among others. In this paper we consider both low- and high-dimensional covariance matrix estimation problems and present a review for asymptotic properties of sample covariances and covariance matrix estimates. In particular, we shall provide an asymptotic theory for estimates of high dimensional covariance matrices in time series, and a consistency result for covariance matrix estimates for estimated parameters. 1 Introduction Covariances and covariance matrices play a fundamental role in the theory and practice of time series. They are critical quantities that are needed in both spectral and time domain analysis. One encounters the issue of covariance matrix estimation in many problems, for example, the construction of confidence regions for unknown parameters, hypothesis testing, principal component analysis, prediction, discriminant analysis among others. It is particularly relevant in time series analysis in which the observations are dependent and the covariance matrix characterizes the second order dependence of the process. If the underlying process is Gaussian, then the covariances completely capture its dependence structure. In this paper we shall provide an asymptotic distributional theory for sample covariances and convergence rates for covariance matrix estimates of time series. In Section 2 we shall present a review for asymptotic theory for sample covariances of stationary processes. -
PROPERTIES of the GAUSSIAN FUNCTION } ) ( { Exp )( C Bx Axy
PROPERTIES OF THE GAUSSIAN FUNCTION The Gaussian in an important 2D function defined as- (x b)2 y(x) a exp{ } c , where a, b, and c are adjustable constants. It has a bell shape with a maximum of y=a occurring at x=b. The first two derivatives of y(x) are- 2a(x b) (x b)2 y'(x) exp{ } c c and 2a (x b)2 y"(x) c 2(x b)2 exp{ ) c2 c Thus the function has zero slope at x=b and an inflection point at x=bsqrt(c/2). Also y(x) is symmetric about x=b. It is our purpose here to look at some of the properties of y(x) and in particular examine the special case known as the probability density function. Karl Gauss first came up with the Gaussian in the early 18 hundreds while studying the binomial coefficient C[n,m]. This coefficient is defined as- n! C[n,m]= m!(n m)! Expanding this definition for constant n yields- 1 1 1 1 C[n,m] n! ... 0!(n 0)! 1!(n 1)! 2!(n 2)! n!(0)! As n gets large the magnitude of the individual terms within the curly bracket take on the value of a Gaussian. Let us demonstrate things for n=10. Here we have- C[10,m]= 1+10+45+120+210+252+210+120+45+10+1=1024=210 These coefficients already lie very close to a Gaussian with a maximum of 252 at m=5. For this discovery, and his numerous other mathematical contributions, Gauss has been honored on the German ten mark note as shown- If you look closely, it shows his curve.