Recent content by Combinatus

Fascinating, thank you! I would have expected to see something more like the bathtub curve that Nugatory mentioned. Interesting derivation of the Weibull distribution's CDF F(𝜏) in the Illuminating physics article (well, 1-F(𝜏), named the surviving fraction). At least the exponential...

I've come across a number of problems in elementary probability theory and statistics that can be exemplified as follows: Naturally, real lamps decay over time, so their lifetimes can't be memoryless. With that being said, is the exponential distribution a good approximation for the...

Well, maybe I'm too tired, but isn't \mathrm{d}z=\dfrac{\mathrm{d}u}{2}?

Substituting u=2z in the first integral yields \displaystyle\int_0^{\infty} \frac{e^{-\dfrac{(x^2+4z^2)}{4z}}}{2z} \, \mathrm{d}z = \displaystyle\int_0^{\infty} \frac{e^{-\dfrac{(x^2+u^2)}{2u}}}{2u} \, \mathrm{d}u. Not bad! I'll try out some more elementary transformations tomorrow. Thanks!

Change of variables for integral Homework Statement Determine whether the following equality holds: \displaystyle\int_0^{\infty} \frac{e^{-\dfrac{(x^2+4z^2)}{4z}}}{2z} \, \mathrm{d}z = \displaystyle\int_0^{\infty} \frac{e^{-\dfrac{(1+z^2)x}{2z}}}{2z} \, \mathrm{d}z, \forall x,z \in...

Thanks for your input! I'm still a bit uncomfortable with u_n = \sin\left(\dfrac{n\pi - \delta}{\pi}x\right) as \delta depends on \lambda, though. I tried a few additional approaches, but I haven't been able to get anything "better" than that.

Homework Statement Show that y''+\lambda y=0 with the initial conditions y(0)=y(\pi)+y'(\pi)=0 has an infinite sequence of eigenfunctions with distinct eigenvalues. Identify the eigenvalues explicitly.Homework Equations The Attempt at a Solution \lambda \le 0 seems to yield the trivial...

I solved it, I think. y = C_1 x^{\frac{1}{2} (1+\sqrt{1-4k})} + C_2 x^{\frac{1}{2} (1-\sqrt{1-4k})} could be written as the composition of sines and cosines of (the monotonic function) \ln x when k > 1/4 with a preceding (monotonic) factor e^{\ln \sqrt{x}}, so y should have infinitely many...

Homework Statement Show that every nontrivial solution of y''+\frac{k}{x^2}y=0 (with k being a constant) has an infinite number of positive zeros if k>1/4 and only finitely many positive zeros if k\le 1/4. Homework Equations The Attempt at a Solution I set y=x^M = e^{M \log{x}} (for some...

Yes, I just noticed that after looking at my scribbles, and hoped no-one would have had time to reply yet. :redface: Oh well. Thanks. :smile:

Homework Statement Find the specified particular solution: (x^2+2y')y'' + 2xy' = 0, y(0)=1, y'(0)=0 Homework Equations The Attempt at a Solution The equation seems amenable to the substitution p=y', so it can be transformed into (x^2 + 2p)p' + 2xp=0, or (x^2 + 2p)dp + 2xpdp=0. Since...

Regular Maclaurin (Taylor at z=0) series for |z|<1, i.e. -1-z-z^2-z^3-\cdots, and \sum_{k = 0}^{\infty} \frac{1}{z^{k+1}} for |z|>1? So, \frac{1}{e^z - 1} = \sum_{k = 0}^{\infty} \frac{1}{e^{z(k+1)}}?

Homework Statement Obtain the first few terms of the Laurent series for the following function in the specified domain: \frac{1}{e^z-1} for 0 < |z| < 2\pi. Homework Equations The Attempt at a Solution I've attempted a few approaches, but haven't really gotten anywhere. For...

Wouldn't it be possible for the limit to be different along some other path? (Although in this particular case, there isn't.) Good find! Using the \left|\sin^2 {\phi} \cos{\theta} \sin{\theta} (1-\cos{(\rho \cos{\phi})}) - \rho \cos^3 {\phi}\right| part from my use of polar coordinates, I...

Homework Statement Find the continuous points P and the differentiable points Q of the function f in {R}^3, defined as f(0,0,0) = 0 and f(x,y,z) = \frac{xy(1-\cos{z})-z^3}{x^2+y^2+z^2}, (x,y,z) \ne (0,0,0). Homework Equations The Attempt at a Solution If you want to look at the limit I'm...