Crash course in Julia (programming)

Julia is a powerful/efficient/high-level computer programming language. You can get into interacting mode right-away with:

julia

You may need to install packages, which can be done as follows:

import Pkg; Pkg.add("Distributions")

Once this is done (once for ever on a given machine), you can then be:

using Distributions

Let us generate ten thousands random points following a squared-uniform probability distribution, $X^2$.

lst=[rand()^2 for i=1:10^5]

and after

using Plots histogram(lst)

The theoretical result is obtained by differentiating the cumulative function, $f_{X^2}=dF_{X^2}/dx$, with

$$F_{X^2}(x)\equiv\mathbb{P}(X^2\le x)$$

but $\mathbb{P}(X^2\le x)=\mathbb{P}(X\le\sqrt{x})=\sqrt{x}$, since the probability is uniform. Therefore:

$$f_{X^2}(x)={1\over2\sqrt{x}}$$

Let us check:

Invalid language.

You need to specify a language like this: <source lang="html4strict">...</source>

Supported languages for syntax highlighting:

4cs, 6502acme, 6502kickass, 6502tasm, 68000devpac, abap, actionscript, actionscript3, ada, algol68, apache, applescript, apt_sources, arm, asm, asp, asymptote, autoconf, autohotkey, autoit, avisynth, awk, bascomavr, bash, basic4gl, bf, bibtex, blitzbasic, bnf, boo, c, c_loadrunner, c_mac, caddcl, cadlisp, cfdg, cfm, chaiscript, cil, clojure, cmake, cobol, coffeescript, cpp, cpp-qt, csharp, css, cuesheet, d, dcl, dcpu16, dcs, delphi, diff, div, dos, dot, e, ecmascript, eiffel, email, epc, erlang, euphoria, f1, falcon, fo, fortran, freebasic, freeswitch, fsharp, gambas, gdb, genero, genie, gettext, glsl, gml, gnuplot, go, groovy, gwbasic, haskell, haxe, hicest, hq9plus, html4strict, html5, icon, idl, ini, inno, intercal, io, j, java, java5, javascript, jquery, kixtart, klonec, klonecpp, latex, lb, ldif, lisp, llvm, locobasic, logtalk, lolcode, lotusformulas, lotusscript, lscript, lsl2, lua, m68k, magiksf, make, mapbasic, matlab, mirc, mmix, modula2, modula3, mpasm, mxml, mysql, nagios, netrexx, newlisp, nsis, oberon2, objc, objeck, ocaml, ocaml-brief, octave, oobas, oorexx, oracle11, oracle8, oxygene, oz, parasail, parigp, pascal, pcre, per, perl, perl6, pf, php, php-brief, pic16, pike, pixelbender, pli, plsql, postgresql, povray, powerbuilder, powershell, proftpd, progress, prolog, properties, providex, purebasic, pycon, pys60, python, q, qbasic, rails, rebol, reg, rexx, robots, rpmspec, rsplus, ruby, sas, scala, scheme, scilab, sdlbasic, smalltalk, smarty, spark, sparql, sql, stonescript, systemverilog, tcl, teraterm, text, thinbasic, tsql, typoscript, unicon, upc, urbi, uscript, vala, vb, vbnet, vedit, verilog, vhdl, vim, visualfoxpro, visualprolog, whitespace, whois, winbatch, xbasic, xml, xorg_conf, xpp, yaml, z80, zxbasic

f(x)=1/(2*sqrt(x))
histogram(lst,norm=true)
plot!(f,.01,1, linewidth = 4, linecolor = :red, linealpha=0.75)

The ! means to plot on the existing display. This seems to work indeed: