Some stuff » PDF

pkfix for those ugly postscript files

admin — Fri, 15 Feb 2013 06:16:26 +0000

This tool, pkfix, is so useful, I cannot even breathe right now. Some old PostScript files generated through the LaTeX/DVI process use bitmap fonts, and when converted to PDF, their texts look ugly or fuzzy, and are unsearchable. People put these up without the source, and it appeared we were out of luck.

Not so! Using pkfix, the bitmap fonts are replaced in the PS file by their equivalent vector font counterparts on the system. Once that’s done, and the PS converted to PDF, the text looks sharp with that freshly typeset look. Search also works.

uniform by three

admin — Sat, 22 Nov 2008 08:08:13 +0000

Here is a problem recently described to me. Apparently there is a more elegant solution (which may give more insight), but I don’t see it yet.

The problem: \(X, Y, Z\) are independent random variables uniformly distributed over [0,1]. What is the distribution of \((XY)^Z\)?

The conventional solution is to find the distribution of \(XY\) first, then of \((XY)^Z\).

The distribution of \(XY\) can be derived from its CDF \(F_{XY}(xy \le k)\), which is the total shaded area shown. The red curve is the function \(y=\frac{k}{x}\). This area is thus given by:

\(\int_k^1 \frac{k}{x} dx + k = k \ln(x) \vert_k^1 + k = -k \ln(k) + k\), for \(k>0\).

The PDF is the derivative of the above:

\(f_{XY}(k) = -\ln(k)\), for \(k>0\). The density is not well defined at \(k=0\).

The second part is to find the distribution in question. Here, the red curve is the function \(z=\frac{\ln(k)}{\ln(xy)}\). The CDF \(F_{(XY)^Z}((xy)^z \le k)\) is the total area to the left of the red curve. A column slice shaded in blue has probability per unit of \(z\) as labeled. Thus, the CDF is:

\(\int_0^k f_{XY}(v) dv (1-\frac{\ln(k)}{\ln(v)}) = \int_0^k -\ln(v) dv (1-\frac{\ln(k)}{\ln(v)})\)
\( = \int_0^k -\ln(v) dv + \int_0^k \ln(k) dv\)
\( = -v \ln(v) + v \vert_{\downarrow 0}^k + k \ln(k) = -k \ln(k) + k + k \ln(k) = k\), for \(k>0\).

For \(k=0\), we can fill in 0, because the minimum value of \((XY)^Z\) is 0, so it is the minimum of the support of its CDF.

So amazingly, \((XY)^Z\) has the CDF of (and is) a uniform distribution. How does that happen? What’s the intuition?

Csiszar & Korner

admin — Mon, 30 Oct 2006 18:37:27 +0000

Imre Csiszar and Janos Korner are two Hungarians with very Hungarian names. But more importantly, they wrote a thrilling page-turner called, Information Theory: Coding Theorems for Discrete Memoryless Systems. It is a book most difficult to obtain. It seems that the book has been out of print ever since the day it was in print. Academiai Kiado of Budapest and Academic Press of New York (same thing?), I’m looking in your general direction(s). Hmm. I wonder if the cost structure of running a printing press is akin to that of running a chip foundry?

Anyway, forget the publishers. There is one copy in the library, permanently checked out, on hold, or requested. Almost never seen in online stores, it sells for several times the list price when scalper123 occasionally trots it out on YahooMazonBay. Worst of all, nobody has bothered to make and distribute a pdf of it for the good of the masses. Er, wait, I mean, nobody has bothered to make a Fair Use copy for personal use.

And accidentally leave the pdf on an unprotected public server. (Please?)

Well, that was last week, and this is now. I am to this day amazed that Kazoo Books still had one (1) old, used, but perfectly good copy at list price. I wrote “had.” Good service and fast delivery, too. No fraud committed against me despite there being a phone transaction with a credit card. Highly recommend. Wait, this isn’t eBay, why am I writing this?