Wikipedia:Reference desk/Archives/Mathematics/2010 August 29

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August 29

linear transformation of standard normal random variables

$\mathbf {Y} =A\mathbf {X}$ where $\mathbf {X}$ is a vector of iid $N(0,1)$ random variables.

I need to show that $\mathbf {Y}$ is also iid $N(0,1)$ if $A$ is an orthogonal matrix.

I have the result that for $\mathbf {Y} =A\mathbf {X}$ we have $f_{\mathbf {Y} }(\mathbf {y} )={\frac {f_{\mathbf {X} }(A^{-1}\mathbf {y} )}{|A|}}$

With an orthogonal matrix substituted into the multivariate standard normal formula things cancel and I get a numerator that works out to be multivariate standard normal which is what I want.

But I'm left with that determinant on the bottom, which for an orthogonal matrix is 1 or -1. If it's 1, no problem. But if it's -1, does that mean my distribution is upside down? —Preceding unsigned comment added by 130.102.158.15 (talk) 02:40, 29 August 2010 (UTC)

Consider first the one-dimensional case. The transformation Y=−X swaps the positive and negative values of X. It does not turn the distribution upside down. Bo Jacoby (talk) 06:41, 29 August 2010 (UTC).

...which means that the denominator should actually be the absolute value of the determinant. -- Meni Rosenfeld (talk) 08:43, 29 August 2010 (UTC)

In this case, if the determinant of the matrix $A$ is 1, I have $f_{y}(y)=f_{x}(y)$ where $f_{x}$ is a normal pdf...but if the determinant is -1, I get $f_{y}(y)=-f_{x}(y)$ and that's not a normal pdf unless $-f_{x}(y)=f_{x}(-y)$, which I'm pretty sure it's not —Preceding unsigned comment added by 130.102.158.15 (talk) 08:44, 29 August 2010 (UTC)

Where did the minus sign come from? -- Meni Rosenfeld (talk) 09:46, 29 August 2010 (UTC)

aaaargh, sorry I had my notation mixed up and didn't realize that the denominator is the ABSOLUTE value of the determinant, I thought it was just the determinant! —Preceding unsigned comment added by 130.102.158.15 (talk) 21:55, 29 August 2010 (UTC)

Euler-Maruyama and Milstein schemes

Can anybody suggest where I might be able to find proofs of the order of convergence for the Euler-Maruyama and/or Milstein schemes used for stochastic differential equations? —Preceding unsigned comment added by Damian Eldridge (talk • contribs) 06:24, 29 August 2010 (UTC)

The Euler-Maruyama method article has a reference that you might look at. 67.119.3.248 (talk) 09:03, 30 August 2010 (UTC)

Trig Question

My days of battling with trig identities are unfortunately a little far behind me. Is it possible to express ${\frac {d}{b^{2}c^{2}-(a^{2}-b^{2})^{2}}}[(a^{2}-b^{2})(\cos(bt))+bc\sin(bt)]$ as $Acos(bt-\theta )$? Thanks asyndeton talk 16:33, 29 August 2010 (UTC)

Actually, maybe they're closer than I think. Is it just $A={\frac {d}{b^{2}c^{2}-(a^{2}-b^{2})^{2}}}$, $tan(\theta )={\frac {a^{2}-b^{2}}{bc}}$? asyndeton talk 16:45, 29 August 2010 (UTC)

Certainly a linear combination of sine and cosine functions of the same period can be expressed as a single cosine function with that same period and with a phase shift. More later...... Michael Hardy (talk) 16:48, 29 August 2010 (UTC)

More specifically, see List_of_trigonometric_identities#Linear_combinations. Michael Hardy (talk) 17:09, 30 August 2010 (UTC)

To quote:

In the case of a linear combination of a sine and cosine wave (which is just a sine wave with a phase shift of π/2), we have

$a\sin x+b\cos x={\sqrt {a^{2}+b^{2}}}\cdot \sin(x+\varphi )\,$

where

$\varphi ={\begin{cases}\arcsin \left({\frac {b}{\sqrt {a^{2}+b^{2}}}}\right)&{\text{if }}a\geq 0,\\\pi -\arcsin \left({\frac {b}{\sqrt {a^{2}+b^{2}}}}\right)&{\text{if }}a<0,\end{cases}}$

or equivalently

$\varphi =\arctan \left({\frac {b}{a}}\right)+{\begin{cases}0&{\text{if }}a\geq 0,\\\pi &{\text{if }}a<0.\end{cases}}$

Michael Hardy (talk) 17:11, 30 August 2010 (UTC)

http://en.wikipedia.org/wiki/Wikipedia:RDMA#Semicircles_problem

here Michael Hardy said that "if a line passes through the centers of two circles in the same plane that touch each other, then it must pass through the point where they touch each other". I can see why this is for externally tangent circles, but why is it true for internally tangent circles as well? in other words, how do you know that the line from the center of the big circle passes through the point the passes through the center and the point of tangency? —Preceding unsigned comment added by 68.248.229.115 (talk) 18:17, 29 August 2010 (UTC)

The circles have a common tangent at the point where they touch. The lines from the centre of each circle to this point meet the tangent line at right angles, hence they coincide. Or just appeal to the symmetry of the layout. →86.132.161.214 (talk) 22:45, 29 August 2010 (UTC)

Or just reflect the inner circle in the common tangent, producing the case of externally tangent circles.→86.132.161.214 (talk) 19:50, 30 August 2010 (UTC)