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- <font size= 3> Topic 10: Characteristic Functions</font size> ==Characteristic Functions==5 KB (804 words) - 12:12, 21 May 2014
- ...expectation E[g(X)], conditional expectation E[g(X)|M], and characteristic function <math>\Phi_X</math>. We will now define similar tools for the case of two r ==Joint Cumulative Distribution Function==8 KB (1,524 words) - 12:12, 21 May 2014
- which is a quadratic function of a ∈ '''R'''. Consider two cases: We have previously seen that the joint density function for Gaussian X and Y contains a parameter r. It can be shown that this r is7 KB (1,307 words) - 12:12, 21 May 2014
- ...sson|Ryan Russon]]: I guess what I am confused about is how I can take the function ''f(x)'' and let ''x''--> ''x+ct'' and ''x-ct'' and then from that, I get t ...lly are getting the superposition of the odd extensions of each 1/2 of the function of x but as it is shifted through time, which in turn just shifts the ''f(x6 KB (1,102 words) - 19:16, 19 November 2013
- * The cumulative distribution function of <span style="text-decoration:underline;">X</span> is <br/> : and the probability density function of <span style="text-decoration:underline;">X</span> is <br/>12 KB (1,897 words) - 12:13, 21 May 2014
- * X(t,<math>\omega_0</math>) is a real-valued function of t for fixed <math>\omega_0</math>. '''Definition''' <math>\qquad</math> The '''autocorrelation function''' of a random process X(t) is <br>10 KB (1,690 words) - 12:13, 21 May 2014
- ...uler's characteristic: X = V - E + F. Any convex polyhedra's surface has a characteristic X = 2, which all five of the Platonic Solids do. ...composition (the application of one function to another to produce a third function) as its operation. Every rotation within the SO(3) group, also has an inver9 KB (1,540 words) - 10:36, 2 December 2013
- Neyman-Pearson Lemma and Receiver Operating Characteristic Curve ...this slecture is understanding Neyman-Pearson Lemma and Receiver Operating Characteristic (ROC) curve from theory to application. The fundamental theories stem from15 KB (2,306 words) - 10:48, 22 January 2015
- According to Euler Characteristic, any convex polyhedron's surface has Euler characteristic E+2=V+F.<br> •Let me use one example to illustrate my understanding of Euler characteristic:<br><br>5 KB (911 words) - 11:27, 27 April 2014
- ''Receiver Operating Characteristic (ROC)'' curve is often used as an important tool to visualize the performan And we use a binary decision rule <math>\ \phi(x)\ </math> as a function:11 KB (1,823 words) - 10:48, 22 January 2015
- (a) Find the (first order) characteristic function of <math class="inline">\mathbf{X}\left(t\right)</math> . (c) Derive an expression for the autocorrelation function of <math class="inline">\mathbf{X}\left(t\right)</math> .4 KB (700 words) - 17:48, 13 March 2015
- which is the characteristic function of a Gaussian random variable with mean <math class="inline">a\mu_{\mathbf{3 KB (525 words) - 00:20, 10 March 2015
- ...class="inline">\left|\rho\right|<1</math> . Find the joint characteristic function <math class="inline">E\left[e^{i\left(h_{1}\mathbf{X}+h_{2}\mathbf{Y}\right Use the fact that the characteristic function of a <math class="inline">N\left(0,1\right)</math> r.v. is given by <math6 KB (895 words) - 00:41, 10 March 2015
- ...math> where <math class="inline">u\left(t\right)</math> is the unit step function and <math class="inline">T>0</math> . What is the autocorrelation function of <math class="inline">\mathbf{Y}\left(t\right)</math> ?6 KB (1,002 words) - 01:38, 10 March 2015
- ...X}</math> is a binomial distributed random variable with probability mass function (pmf) given by <math class="inline">p_{n}\left(k\right)=\left(\begin{array} ...line">\mathbf{X}</math> . (You must show how you derive the characteristic function.)4 KB (758 words) - 01:56, 10 March 2015
- ..., with <math class="inline">\mathbf{X}_{n}</math> having probability mass function <math class="inline">p_{n}\left(k\right)=\left(\begin{array}{c} ...nline">e^{\lambda\left(e^{i\omega}-1\right)}</math> is the characteristic function of a Poisson random variable with mean <math class="inline">\lambda</math>3 KB (429 words) - 01:57, 10 March 2015
- Find the characteristic function of <math class="inline">\mathbf{X}_{n}</math> . ...t(1-\left(\frac{1}{2}\right)^{2n}\right)</math> . Hence the characteristic function of <math class="inline">\mathbf{X}_{n}</math> is <math class="inline">\Phi4 KB (692 words) - 10:36, 10 March 2015
- ...function <math class="inline">\mu\left(t\right)</math> and autocovariance function <math class="inline">C_{\mathbf{XX}}\left(t_{1},t_{2}\right)</math> . ...expression for the <math class="inline">n</math> -th order characteristic function of <math class="inline">\mathbf{X}\left(t\right)</math> in terms of <math3 KB (451 words) - 10:56, 10 March 2015
- ...nverge in distribution]]. Furthermore, you have to know the characteristic function of Cauchy distributed random varaible. According to the characteristic function of Cauchy distributed random variable,3 KB (451 words) - 10:58, 10 March 2015
- Then, compute the characteristic function of <math>\mathbf{X}+\mathbf{Y}</math>: We know that the characteristic function of a Gaussian random variable with mean <math>\mu</math> and variance <math5 KB (882 words) - 01:54, 31 March 2015
