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  1. L

    To show that an integral is divergent

    Thanks for your reply. That's what i thought. So I assume the easiest/typical way is the way I described above ?
  2. L

    To show that an integral is divergent

    Hi all I am looking for a simple way to show that the mean of the Cauchy distribution us undefined. This is because this integral diverges: \underset{-\infty}{\overset{\infty}{\int}}\frac{x}{x^{2}+a^{2}}dx Now, I know one proof which replaces the limits of integration with -x1 and x2. After...
  3. L

    Please convince me (OLS matrix derivation)

    OK, here goes again. I think I fixed the LaTex problems: let \mathbf{A=X^{\textrm{T}}X} where \mathbf{A} is a n\times n square symmetric matrix with elements a_{ij}. \mathbf{\mathbf{\boldsymbol{\beta}}} is the n\times 1 column vector. Expanding out...
  4. L

    LaTeX LaTex advice/help

    Hi Fredrik Thanks so much for the explanation ! So, just to be clear if I put begin{align} end{align} around the text from Lyx, it should solve the problem ? Thanks again LR
  5. L

    LaTeX LaTex advice/help

    Hi all I use Lyx v1.65 for creating LaTex. Obviously it displays OK in Lyx. It also displays OK with an online editor I've tried http://www.codecogs.com/latex/eqneditor.php But invariably when I try to post LaTex to this forum it does not work. I've tried the sandbox which the reason...
  6. L

    Please convince me (OLS matrix derivation)

    Hi again I need to stick with the pure linear algebraic derivation at the moment, but thanks anyway. I may come back to you later on that, as I am also interested in the geometric interpretation. Anyway, I think I have solved it. Basically, it revolves around the "rule" that the derivative...
  7. L

    Please convince me (OLS matrix derivation)

    Hi Bacle, thanks for your messages. I'm glad I'm not the only one who is a bit confused by it. For completeness and the benefit of others, I'll explain the setup so that it's not necessary to refer to the link I posted. We have y = XB + e where y is a n x 1 column vector of responses X is a...
  8. L

    Please convince me (OLS matrix derivation)

    Excuse me: I meant "the 3rd term in the expression above" not "the 2nd term on the RHS".
  9. L

    Please convince me (OLS matrix derivation)

    Hi and thanks for your reply. Could you take a look here: http://cran.r-project.org/doc/contrib/Faraway-PRA.pdf On page 18/19 you see exactly what (I think) you are referring to in terms of the orthogonal projection. What I am referring to is on the bottom of page 19: "Differentiating with...
  10. L

    Please convince me (OLS matrix derivation)

    Hi all In the derivation of the normal equations for Ordinary Least Squares estimates we have B (m x 1 column vector) and X (n x m matrix). Could someone please convince me that the derivative with respect to B of B'X'XB is 2X'XB Thanks ! LR
  11. L

    Probability density: change of variable

    If x is a random variable uniformly continuously distributed on [0.1], and y=x^3, then y has the density: \frac{1}{3}y^{-2/3} on [0,1] But, if x has the same distribution, but on [-0.5, 0.5], there seems to be a problem because we have y^{-2/3} for negative values of y. This is overcome if we...
  12. L

    A textbook on probability and statistics

    I like Probability and Random Processes, Third Edition, by Grimmett and Stirzacker, (Oxford University Press, 2001).
  13. L

    Obvious integral manipulation

    This wasn't obvious to me. From my book. We have, p(x)=\begin{cases} \frac{1}{2} & -2\leq x\leq-1\,\textrm{or}\;1\leq x\leq2,\\ 0 & \text{otherwise~}.\end{cases} So, the kth moment is given by M_{k}=\frac{1}{2}\int_{-2}^{-1}x^{k}dx+\frac{1}{2}\int_{1}^{2}x^{k}dx So, obviously...
  14. L

    Weibull integral

    Thanks !
  15. L

    Weibull integral

    Can someone explain this.. P(v)=\frac{\beta}{\eta}\intop_{0}^{v}\left(\frac{v}{\eta}\right)^{\beta-1}\exp\left(-\left(\frac{v}{\eta}\right)^{\beta}\right)dv=\intop_{0}^{x}e^{-x}dx\hphantom{}\; where\phantom{\:}x=\left(\frac{v}{\eta}\right)^{\beta} Thanks !
  16. L

    Evaluation of limit

    Got it. Thanks again.
  17. L

    Evaluation of limit

    Dick, Your final remark about the more formal approach requires me to know the answer, ie g(x) already, which was obtained by the "casual" approach. So how do you find g(x) more formally than using the casual approach ? Thanks a lot !
  18. L

    Evaluation of limit

    Thank you very much.
  19. L

    Maple Maple indentation

    I don't think you can, I just use 2 spaces for 1 indent.
  20. L

    Evaluation of limit

    in case I'm not being clear, the expression I posted initially is the general solution to a DE problem that I obtained (so it might be wrong !) and the question is to show that the long term behaviour as x->infinitty is that x approaches (1/2)e^(-x)
  21. L

    Evaluation of limit

    The question doesn't explicitly ask for the limit, I just assumed that's what I had to do. This is an applied maths course, not analysis. The question is asking for the long term behaviour of the solution to a differential equation...
  22. L

    Evaluation of limit

    Because I am told the answer is \frac{1}{2}e^{-x}
  23. L

    Evaluation of limit

    Homework Statement Evaluate the limit of the following as x approaches infinity \frac{e^{x}-1}{1-2e^{x}+2e^{2x}} Homework Equations The Attempt at a Solution \frac{e^{-x}-e^{-2x}}{e^{-2x}-2e^{-x}+2} which gives 0/2=0 as x approaches infinity, but apparently this is wrong.
  24. L

    Amplitude of harmonic oscillator

    Indeed, yes. So easy now. Thanks !
  25. L

    Amplitude of harmonic oscillator

    In Maple: {\it s2s}\, := \,{\frac { \left( -{\omega_{{0}}}^{2}+{\omega}^{2} \right) \sin \left( \omega\,t \right) +\cos \left( \omega\,t \right) \omega\,\lambda}{{\omega}^{4}+ \left( {\lambda}^{2}-2\,{\omega_{{0}}}^{2} \right) {\omega}^{2}\\ \mbox{}+{\omega_{{0}}}^{4}}}\]} which simplifies to...
  26. L

    Amplitude of harmonic oscillator

    Another look at this in Maple: [Sorry, fiddling with latex...]
  27. L

    Amplitude of harmonic oscillator

    Yes I'm sure it's correct. This is homework so I'm loathed to post the whole details. However, take a look here: http://nuweb.neu.edu/dheiman/U600/DHO.pdf Look at the section "Driven Harmonic Oscillator" on page 4. The solution given is exactly what I want (with lambda, w_0 and F_0/m all equal...
  28. L

    Amplitude of harmonic oscillator

    But the coefficients of each sinusoidal term are different...
  29. L

    Amplitude of harmonic oscillator

    Homework Statement I have a simple harmonic oscillator system with the driving force a sinusoidal term. The question is to find the general solution and the amplitude of the steady state solution Homework Equations I found the steady state part of the solution. It is of the form...
  30. L

    Need help with Differential equation

    James' equation for y is just the (explicit) general solution which follows from the solution of the characteristic equation (repeated real root). His equation for y' should have just been the derivative but he forgot to use the product rule for the term involving the constant B. Sorry, but I...
  31. L

    Implicit differentiation

    OK, more on this. Finally I think I have it. Both methods result in -y/x. In the simple method where we cancel x first, we have the proviso that x is not equal to 0 and in the method I wrote initially, my "result" also relies on x not equal to 0 (and also cos(xy) not equal to zero). Once this...
  32. L

    Need help with Differential equation

    Cheer up :) It's an easy mistake to make. The main point is that you got the main method right. If I had a pound for the number of times I've made mistakes like that I'd be very rich ! See my thread on implicit differentiation if you want a laugh !
  33. L

    Need help with Differential equation

    Pretty sure... No, the derivative of A.exp(-5t) is -5A.exp(-5t) and the derivative of Bt.exp(-5t) is -5Bt.exp(-5t) + B.exp(-5t)
  34. L

    Implicit differentiation

    Sorry, I am still not sure on this... Is it really valid to cancel x here ? How did cos(xy) disappear ? And why is my attempt in my initial post wrong ?
  35. L

    Need help with Differential equation

    you need to differentiate the GS and plug y=1 into it. y'=B.exp(-5t) - 5(A+Bt).exp(-5t) Since A+B=0,y'(1)=B.exp(-5)=2 so B=2.exp(5) and A=-2.exp(5)
  36. L

    Implicit differentiation

    OK, I've got it now. Seems I've been led astray by the "apparent" answer.
  37. L

    Implicit differentiation

    The "apparent" answer came from http://www.analyzemath.com/calculus/Differentiation/implicit.html See the bottom of the page If the actual answer is dy/dx = -y/x, please can you explain. I showed my working in my initial posting. Where did I go wrong ?
  38. L

    Linear Equations

    Of course, my method is the same as hisham's but I just find it easier to follow.
  39. L

    Linear Equations

    This is the method I use: Take a generic linear 1st order ODE: y' + g(x)y = h(x) Then, the integrating factor I(x) = exp(int(g(x) dx)) After multiplying through by I(x) we can write: (I(x)y)' = I(x)h(x) because the integrating factor always has the property that after multiplication the LHS...
  40. L

    Implicit differentiation

    I can easily see that dy/dx=-y/x when sin(xy)=0, so I'm not sure how that helps, but thanks anyway !
  41. L

    Linear Equations

    The first thing you need to do is express the DE in "standard form" so that nothing is multiplying y'...so in your last example you need to multiply through by x, so it becomes y'+y/x = x. Not sure where your last equation comes from - maybe you made some typos ? Then the integrating factor will...
  42. L

    Linear Equations

    That looks wrong to me. Surely a(x)=I(x)=x^{-2}, giving a solution of y(x)=x^{2}(\ln(x)+C)
  43. L

    Linear Equations

    What do you have for I(x) ? For writing latex, click the sigma symbol on the toolbar, but I found it easier to get started using something like this http://www.codecogs.com/components/equationeditor/equationeditor.php [Broken]
  44. L

    Implicit differentiation

    Homework Statement find \frac{\mathrm{d}y}{\mathrm{d}x} where y is defined implicitly as a function of x Homework Equations x\sin(xy)=x The Attempt at a Solution x(\cos(xy)(x\frac{\mathrm{d}y}{\mathrm{d}x}+y))+\sin(xy)=1...
  45. L

    Moment of inertia of a lamina

    Thank you. That makes perfect sense !
  46. L

    Moment of inertia of a lamina

    Homework Statement A lamina of unit density consists of the region between the two curves y=\sqrt{4-x^2} and y=1-4x^2 and the x axis. Find it's moment of inertia about the x-axis. Homework Equations This is the correct answer: 2\left \{ \int_{0}^{2}\int_{0}^{\sqrt{4-x^2}}}y^2 dy dx...
  47. L

    What's this function ?

    Perfect. Thank you.
  48. L

    What's this function ?

    Thanks. I considered arctan already, but since this function goes momentarily vertical zero arctan doesn't work. Same with a Gompertz function and Richards curve (I think). Also, this function appears to be odd, so that would rule out a Gompterz function also. Are there Sigmoid curves that are...
  49. L

    What's this function ?

    Thanks, but I've not been able to find one. Any suggestions ?
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